Tilley R.J.D. Colour and the Optical Properties of Materials (Wiley, 2011

Home , Lithium iodate, Lithium triborate

Colour and the Optical Properties of Materials

An Exploration of the Relationship Between Light, the Optical Properties of Materials and Colour

PROFESSOR RICHARD J. D. TILLEY Emeritus Professor, University of Cardiff, UK

Colour and the Optical Properties of Materials

An Exploration of the Relationship Between Light, the Optical Properties of Materials and Colour

PROFESSOR RICHARD J. D. TILLEY Emeritus Professor, University of Cardiff, UK This edition first published 2011 Ó 2011 John Wiley & Sons, Ltd Registered office John Wiley & Sons, Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, withoutthe prior permission of the publisher.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.

Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. The publisher and the author make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of fitness for a particular purpose. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for every situation. In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. The fact that an organization or Website is referred to in this work as a citation and/or a potential source of further information does not mean that the author or the publisher endorses the information the organization or Website may provide or recommendations it may make. Further, readers should be aware that Internet Websites listed in this work may have changed or disappeared between when this work was written and when it is read. No warranty may be created or extended by any promotional statements for this work. Neither the publisher nor the author shall be liable for any damages arising herefrom.

Library of Congress Cataloging-in-Publication Data

Tilley, R. J. D. Colour and the optical properties of materials : an exploration of the relationship between light, the optical properties of materials and colour / Richard J. D. Tilley. p. cm. Includes bibliographical references and index. ISBN 978-0-470-74696-7 (cloth) – ISBN 978-0-470-74695-0 (pbk.) 1. Light. 2. Optics. 3. Color. I. Title. QC355.3.T55 2010 535.6–dc22 2010025108

A catalogue record for this book is available from the British Library. ISBN 9780470746967 [HB] ISBN 9780470746950 [PB]

Set in 10/12pt Times Roman by Thomson Digital, Noida, India To Anne, for her continued help and support

Contents

Preface xv

1 Light and Colour 1 1.1 Colour and Light 1 1.2 Colour and Energy 3 1.3 Light Waves 5 1.4 Interference 7 1.5 Light Waves and Colour 9 1.6 Black-Body Radiation and Incandescence 10 1.7 The Colour of Incandescent Objects 13 1.8 Photons 14 1.9 Lamps and Lasers 16 1.9.1 Lamps 16 1.9.2 Emission and Absorption of Radiation 17 1.9.3 Energy-Level Populations 17 1.9.4 Rates of Absorption and Emission 18 1.9.5 Cavity Modes 21 1.10 Vision 23 1.11 Colour Perception 28 1.12 Additive Coloration 29 1.13 The Interaction of Light with a Material 33 1.14 Subtractive Coloration 37 1.15 Electronic ‘Paper’ 39 1.16 Appearance and Transparency 40 Appendix A1.1 Deﬁnitions, Units and Conversion Factors 43 A1.1.1 Constants, Conversion Factors and Energy 43 A1.1.2 Waves 43 A1.1.3 SI Units Associated with Radiation and Light 45 Further Reading 47

2 Colours Due to Refraction and Dispersion 49 2.1 Refraction and the Refractive Index of a Material 49 2.2 Total Internal Reﬂection 54 2.2.1 Refraction at an Interface 54 Contents viii

2.2.2 Evanescent Waves 54 2.3 Refractive Index and Polarisability 58 2.4 Refractive Index and Density 60 2.5 Invisible Animals, GRINs and Mirages 62 2.6 Dispersion and Colours Produced by Dispersion 65 2.7 Rainbows 68 2.8 Halos 75 2.9 Fibre Optics 75 2.9.1 Optical Communications 75 2.9.2 Optical Fibres 77 2.9.3 Attenuation in Glass Fibres 79 2.9.4 Chemical Impurities 80 2.9.5 Dispersion and Optical-Fibre Design 81 2.10 Negative Refractive Index Materials 84 2.10.1 Metamaterials 84 2.10.2 Superlenses 87 Further Reading 89

3 The Production of Colour by Reflection 91 3.1 Reflection from a Single Surface 92 3.1.1 Reflection from a Transparent Plate 92 3.1.2 Data Storage Using Reflection 94 3.2 Interference at a Single Thin Film in Air 94 3.2.1 Reflection Perpendicular to the Film 96 3.2.2 Variation with Viewing Angle 97 3.2.3 Transmitted Beams 98 3.3 The Colour of a Single Thin Film in Air 99 3.4 The Reflectivity of a Single Thin Film in Air 101 3.5 The Colour of a Single Thin Film on a Substrate 102 3.6 The Reflectivity of a Single Thin Film on a Substrate 104 3.7 Low-Reflection and High-Reflection Films 105 3.7.1 Antireflection Coatings 105 3.7.2 Antireflection Layers 106 3.7.3 Graded Index Antireflection Coatings 108 3.7.4 High-Reflectivity Surfaces 110 3.7.5 Interference-Modulated (IMOD) Displays 110 3.8 Multiple Thin Films 111 3.8.1 Dielectric Mirrors 111 3.8.2 Multilayer Stacks 113 3.8.3 Interference Filters and Distributed Bragg Reflectors 114 3.9 Fibre Bragg Gratings 115 3.10 ‘Smart’ Windows 119 3.10.1 Low-Emissivity Windows 119 3.10.2 Self-Cleaning Windows 121 3.11 Photonic Engineering in Nature 121 3.11.1 The Colour of Blue Butterflies 122 3.11.2 Shells 122 ix Contents

3.11.3 Labradorite 122 3.11.4 Mirror Eyes 125 Appendix A3.1 The Colour of a Thin Film in White Light 126 Further Reading 127

4 Polarisation and Crystals 129 4.1 Polarisation of Light 129 4.2 Polarisation by Reﬂection 131 4.3 Polars 135 4.4 Crystal Symmetry and Refractive Index 137 4.5 Double Refraction: Calcite as an Example 138 4.5.1 Double Refraction 138 4.5.2 Refractive Index and Crystal Structure 140 4.6 The Description of Double Refraction Effects 143 4.6.1 Uniaxial Crystals 143 4.6.2 Biaxial Crystals 144 4.7 Colour Produced by Polarisation and Birefringence 147 4.8 Dichroism and Pleochroism 149 4.9 Nonlinear Effects 151 4.9.1 Nonlinear Crystals 151 4.9.2 Second- and Third-Harmonic Generation 153 4.9.3 Frequency Mixing 155 4.9.4 Optical Parametric Ampliﬁers and Oscillators 156 4.10 Frequency Matching and Phase Matching 157 4.11 More on Second-Harmonic Generation 160 4.11.1 Polycrystalline Solids and Powders 160 4.11.2 Second-Harmonic Generation in Glass 160 4.11.3 Second-Harmonic and Sum-Frequency-Generation by 161 Organic Materials 4.11.4 Second-Harmonic Generation at Interfaces 162 4.11.5 Second-Harmonic Microscopy 162 4.12 Optical Activity 162 4.12.1 The Rotation of Polarised Light 162 4.12.2 Circular Birefringence and Dichroism 166 4.13 Liquid Crystals 168 4.13.1 Liquid-Crystal Mesophases 168 4.13.2 Liquid-Crystal Displays 169 Further Reading 173

5 Colour Due to Scattering 175 5.1 Scattering and Extinction 175 5.2 Tyndall Blue and Rayleigh Scattering 176 5.3 Blue Skies, Red Sunsets 178 5.4 Scattering and Polarisation 181 5.5 Mie Scattering 184 5.6 Blue Eyes, Blue Feathers and Blue Moons 187 5.7 Paints, Sunscreens and Related Matters 188 Contents x

5.8 Multiple Scattering 190 5.9 Gold Sols and Ruby Glass 191 5.10 The Lycurgus Cup and Other Stained Glass 193 Further Reading 195

6 Colour Due to Diffraction 197 6.1 Diffraction and Colour Production by a Slit 198 6.2 Diffraction and Colour Production by a Rectangular Aperture 200 6.3 Diffraction and Colour Production by a Circular Aperture 202 6.4 The Diffraction Limit of Optical Instruments 203 6.5 Colour Production by Linear Diffraction Gratings 205 6.6 Two-Dimensional Gratings 208 6.7 Estimation of the Wavelength of Light by Diffraction 210 6.8 Diffraction by Crystals and Crystal-like Structures 211 6.8.1 Bragg’s Law 211 6.8.2 Opals 213 6.8.3 Artificial and Inverse Opals 218 6.8.4 The Effective Refractive Index of Inverse Opals 221 6.8.5 Photonic Crystals and Photonic Band Gaps 223 6.8.6 Dynamical Form of Bragg’s Law 224 6.9 Diffraction from Disordered Gratings 225 6.9.1 Random Specks and Droplets 225 6.9.2 Colour from Cholesteric Liquid Crystals 228 6.9.3 Disordered Two- and Three-Dimensional Gratings 230 6.10 Diffraction by Sub-Wavelength Structures 231 6.10.1 Diffraction by Moth-Eye Antireflection Structures 231 6.10.2 The Cornea of the Eye 233 6.10.3 Some Blue Feathers 234 6.11 Holograms 235 6.11.1 Holograms and Interference Patterns 235 6.11.2 Transmission Holograms 235 6.11.3 Reflection Holograms 237 6.11.4 Rainbow Holograms 239 6.11.5 Hologram Recording Media 240 6.11.6 Embossed Holograms 242 Further Reading 243

7 Colour from Atoms and Ions 247 7.1 The Spectra of Atoms and Ions 247 7.2 Terms and Levels 252 7.3 Atomic Spectra and Chemical Analysis 254 7.4 Fraunhofer Lines and Stellar Spectra 255 7.5 Neon Signs and Early Plasma Displays 256 7.6 The Helium Neon Laser 259 7.7 Sodium and Mercury Street Lights 262 7.8 Transition Metals and Crystal-Field Colours 264 7.9 Crystal Field Splitting, Energy Levels and Terms 270 xi Contents

7.9.1 Configurations and Strong Field Energy Levels 270 7.9.2 Weak Fields and Term Splitting 271 7.9.3 Intermediate Fields 273 7.10 The Colour of Ruby 277 7.11 Transition-Metal-Ion Lasers 281 7.11.1 The Ruby Laser: A Three-Level Laser 281 7.11.2 The Titanium Sapphire Laser 282 7.12 Emerald, Alexandrite and Crystal-Field Strength 283 7.13 Crystal-Field Colours in Minerals and Gemstones 284 7.14 Colour as a Structural Probe 287 7.15 Colours from Lanthanoid Ions 288 7.16 The Neodymium (Nd3+) Solid-State Laser: A Four-Level Laser 290 7.17 Amplification of Optical-Fibre Signals 294 7.18 Transition Metal, Lanthanoid and Actinoid Pigments 295 7.19 Spectral-Hole Formation 297 Appendix A7.1 Electron Configurations 300 A7.1.1 Electron Configurations of the Lighter Atoms 300 A7.1.2 The 3d Transition Metals 301 A7.1.3 The Lanthanoid (Rare Earth) Elements 301 Appendix A7.2 Terms and Levels 302 A7.2.1 The Vector Model of the Atom 302 A7.2.2 Energy Levels and Terms of Many-Electron Atoms 304 A7.2.3 The Ground-State Term of an Atom 306 A7.2.4 Energy Levels of Many-Electron Atoms 306 Further Reading 307

8 Colour from Molecules 309 8.1 The Energy Levels of Molecules 309 8.2 The Colours Arising in Some Simple Inorganic Molecules 311 8.3 The Colour of Water 315 8.4 Chromophores, Chromogens and Auxochromes 316 8.5 Conjugated Bonds in Organic Molecules: The Carotenoids 317 8.6 Conjugated Bonds Circling Metal Atoms: Porphyrins and Phthalocyanines 319 8.7 Naturally Occurring Colorants: Flavonoid Pigments 323 8.7.1 Flavone-Related Colours: Yellows 323 8.7.2 Anthocyanin-Related Colours: Reds and Blues 324 8.7.3 The Colour of Red Wine 328 8.8 Autumn Leaves 332 8.9 Some Dyes and Pigments 333 8.9.1 Indigo, Tyrian Purple and Mauve 335 8.9.2 Tannins 337 8.9.3 Melanins 337 8.10 Charge-Transfer Colours 340 8.10.1 Charge-Transfer Processes 340 8.10.2 Cation-to-Cation (Intervalence) Charge Transfer 341 8.10.3 Anion-to-Cation Charge Transfer 345 8.10.4 Iron-Containing Minerals 346 Contents xii

8.10.5 Intra-Anion Charge Transfer 348 8.11 Colour-Change Sensors 349 8.11.1 The Detection of Metal Ions 349 8.11.2 Indicators 350 8.11.3 Colorimetric Sensor Films and Arrays 353 8.11.4 Markers 354 8.12 Dye Lasers 355 8.13 Photochromic Organic Molecules 358 Further Reading 361

9 Luminescence 363 9.1 Luminescence 363 9.2 Activators, Sensitisers and Fluorophores 365 9.3 Atomic Processes in Photoluminescence 368 9.3.1 Energy Absorption and Emission 368 9.3.2 Kinetic Factors 370 9.3.3 Quantum Yield and Reaction Rates 371 9.3.4 Structural Interactions 374 9.3.5 Quenching 374 9.4 Fluorescent Lamps 379 9.4.1 Halophosphate Lamps 379 9.4.2 Trichromatic Lamps 381 9.4.3 Other Fluorescent Lamps 382 9.5 Plasma Displays 383 9.6 Cathodoluminescence and Cathode Ray Tubes 385 9.6.1 Cathode Rays 385 9.6.2 Television Tubes 386 9.6.3 Other Applications of Cathodoluminescence 389 9.7 Field-Emission Displays 390 9.8 Phosphor Electroluminescent Displays 391 9.9 Up-Conversion 394 9.9.1 Ground-State Absorption and Excited-State Absorption 395 9.9.2 Energy Transfer 399 9.9.3 Other Up-Conversion Processes 401 9.10 Quantum Cutting 402 9.11 Fluorescent Molecules 405 9.11.1 Molecular Fluorescence 405 9.11.2 Fluorescent Proteins 407 9.11.3 Fluorescence Microscopy 409 9.11.4 Multiphoton Excitation Microscopy 410 9.12 Fluorescent Nanoparticles 411 9.13 Fluorescent Markers and Sensors 412 9.14 Chemiluminescence and Bioluminescence 413 9.15 Triboluminescence 416 9.16 Scintillators 416 Further Reading 418 xiii Contents

10 Colour in Metals, Semiconductors and Insulators 419 10.1 The Colours of Insulators 420 10.2 Excitons 421 10.3 Impurity Colours in Insulators 424 10.4 Impurity Colours in Diamond 424 10.5 Colour Centres 429 10.5.1 The F Centre 429 10.5.2 Electron and Hole Centres 430 10.5.3 Surface Colour Centres 434 10.5.4 Complex Colour Centres: Laser Action 434 10.5.5 Photostimulable Phosphors 435 10.6 The Colours of Inorganic Semiconductors 436 10.6.1 Coloured Semiconductors 436 10.6.2 Transparent Conducting Oxides 437 10.7 The Colours of Semiconductor Alloys 440 10.8 Light Emitting Diodes 441 10.8.1 Direct and Indirect Band Gaps 441 10.8.2 Idealised Diode Structure 443 10.8.3 High-Brightness LEDs 445 10.8.4 Impurity Doping in LEDs 446 10.8.5 LED Displays and White Light Generation 446 10.9 Semiconductor Diode Lasers 448 10.10 Semiconductor Nanostructures 449 10.10.1 Nanostructures 449 10.10.2 Quantum Wells 451 10.10.3 Quantum Wires and Quantum Dots 454 10.11 Organic Semiconductors and Electroluminescence 457 10.11.1 Molecular Electroluminescence 457 10.11.2 Organic Light Emitting Diodes 459 10.12 Electrochromic Films 463 10.12.1 Tungsten Trioxide Electrochromic Films 465 10.12.2 Inorganic Electrochromic Materials 467 10.12.3 Electrochromic Molecules 468 10.12.4 Electrochromic Polymers 468 10.13 Photovoltaics 471 10.13.1 Photoconductivity and Photovoltaic Solar Cells 471 10.13.2 Dye-Sensitised Solar Cells 472 10.14 Digital Photography 474 10.14.1 Charge Coupled Devices 474 10.14.2 CCD Photography 476 10.15 The Colours of Metals 477 10.16 The Colours of Metal Nanoparticles 478 10.16.1 Plasmons 478 10.16.2 Surface Plasmons and Polaritons 479 10.16.3 Polychromic Glass 481 10.16.4 Photochromic Glass 482 10.16.5 Photographic Film 484 Contents xiv

10.16.6 Metal Nanoparticle Sensors and SERS 486 10.17 Extraordinary Light Transmission and Plasmonic Crystals 487 Further Reading 488 Index 491 Preface

This book is concerned with colour. It is not primarily a textbook on optics, but focuses attention upon the ways that colour can be produced and how these ways govern device applications. However it is not possible to discuss colour without reference to numerous optical properties, so these, too, are explained throughout the text. Colour, though, remains the dominant theme. When writing about colour and colour production from a scientific point of view one is beset by a number of language conflicts, arising from the historical importance of the subject. Much of this confusion is due to the fact that the terminology has arisen gradually, as a result of historical experiences that scientists of the day found difficult to understand and interpret. Thus, diffraction, scattering, reflection and refraction can all be considered to be scattering of photons, and the variety of terms in use only confuses the modern reader. Indeed, the nature of light itself leads to problems. Is it a series of waves or a spray of bullet-like photons? A light wave can apparently pass through holes in a metal foil that are far smaller than its wavelength. How can this be? Is the light, instead, a series of photons that can do this, and if so, how big is a photon? Other similar difficulties exist. A decaying and glowing fungus exhibiting bioluminescence does not produce light by the same mechanism as a light-emitting diode (LED) using electroluminescence, although both are termed luminescence. The termination ‘-chromic’ suffers from the same lack of precision. Thermochromic molecules may or may not exhibit colour changes by the same mechanism as electrochromic thin films. The names do not supply any information about this. The units used in the measurement of light are equally confusing. This is because absolute measurements of energy, radiometric units, do not correspond to visual perception, measured in photometric units. Many of these questions are resolved in this book, particularly with respect to light and colour. The explanations are taken at as simple a level that will allow an appreciation of the topic. The book falls into three recognizable sections. Chapter 1 is introductory and covers ideas of light as rays, waves or photons. The emission and absorption of radiation is described, as is the difference in light from an incandescent source and light from a laser. Vision and the perception of colour (physiology or psychology), and related aspects are described in outline, as is the technical measurement of colour. These are specialist topics, and the information here is designed only to cover the need of subsequent chapters. Finally, the way in which light can interact with a material is summarized, as a prologue to later chapters. Chapters 2 6 explain optical phenomena mostly in terms of light waves. Colour is generated when light waves comprising all colours (white light) are subdivided physically into a series of smaller wavelength ranges (i.e. colours). Traditional divisions of the topic are retained, although there is little to choose, theoretically, between labelling a process scattering, diffraction or reflection. Because of this, there is sometimes an ambiguity as to where a particular topic should be placed. For example, fibre Bragg gratings might be treated as multiple reflectors or as diffraction arrays. Mie scattering can be regarded as diffraction. The layout adopted here is one that fits best with the explanations involved. Preface xvi

Chapters 7 10 require a photon explanation to account for colour production. Fundamentally, the absorption and emission of light from atoms, ions and molecules forms the central theme, and for this a quantum mechanical approach is needed. Many of these processes are widely exploited in displays. Although these are technologically complex and require considerable engineering skills in production, the way in which they produce colour is always based upon recognisable physical and chemical principles. Because of this, displays are introduced throughout the text in terms of the appropriate colour-generating mechanism, rather than as a separate section. The topics covered encroach upon physics, chemistry, biology, materials science and engineering and many aspects of these intertwined subject areas are touched upon. Students of all of these disciplines should ﬁnd this book of relevance to some of their studies or interests. Readers who need more information can turn to the Further Reading sections at the end of each chapter. These include selected references to the original literature or substantial reviews and will allow them to take matters further. In addition, the website (www.wiley.com/go/ tilley colour) that accompanies this book contains exercises and numerical problems which have been provided to illustrate and reinforce the concepts presented in the text. All readers are encouraged to attempt them. There are also introductory questions that appear at the start of each chapter which are designed to stimulate interest. The answers to these are found in the Chapter itself. In addition, the answers to these introductory questions and all the other exercises and problems are to be found on the accompanying website. Unfortunately, some important light-related topics have been omitted. These include the important biological topics of photoperiodism inanimals and plants and photosynthesis. Although colour is ofimportance in these topics, the specialist knowledge here is biological rather than optical, and information in this ﬁeld is best reserved for biological texts. It is a pleasure to acknowledge the considerable help and encouragement received in the preparation of this edition. The editorial staff of John Wiley & Sons have always given both assistance and encouragement in the venture. I am indebted to Professor D. J. Brown, University of California, Irvine, USA; Mr A. Dulley, West Glamorgan Archive Service, Swansea, Wales; Dr A. Eddington, Dr J. A. Findlay; Professor I. C. Freestone, University of Cardiff, Wales; Spectrum Technologies plc, Bridgend, Wales; Dr M. Sugdon, De La Rue Group; Dr R. D. Tilley, Victoria University of Wellington, New Zealand; Professor X. Zhang, University of California, Berkeley, USA; Dr P.Vukusic, University of Exeter, England; Dr G. I. N. Waterhouse, University of Auckland, New Zealand. All of them readily provided photographic material. To all of these I express my sincere thanks. Allan Coughlin gave encouragement and advice, and the members of staff of the Trevithick Library, University of Cardiff, Wales, were indefatigable in answering my obscure queries. Finally, my thanks, as always, are due to my wife Anne, who tolerated my hours reading or sat in front of a computer without complaint, and made it possible to complete this work.

Richard J. D. Tilley South Glamorgan May 2010 1 Light and Colour

. What is colour? . Why do hot objects become red or white hot? . How do e-books produce ‘printed’ words?

1.1 Colour and Light

Colour is defined as the subjective appearance of light as detected by the eye. It is necessary, therefore, to look initially at how light is regarded. In fact, light has been a puzzle from earliest times and remains so today. In elementary optics, light can usefully be considered to consist of light rays. These can be thought of as extremely fine beams that travel in straight lines from the light source and thence, ultimately, to the eye. The majority of optical instruments can be constructed within the framework of this idea. However, the ray concept breaks down when the behaviour of light is critically tested, and the performance of optical instruments, as distinct from their construction, cannot be explained in terms of light rays. Moreover, colour is not conveniently defined in this way. For this, more complex ideas are needed. The first testable theory of the nature of light was put forward by Newton (in 1704) in his book Optics,in which it was suggested that light was composed of small particles or ‘corpuscles’. This idea was supported on philosophical grounds by Descartes. Huygens, a contemporary, thought that light was wavelike, a point of view also supported by Hooke. Young provided strong evidence for the wave theory of light by demonstrating the interference of light beams (1803). Shortly afterwards, Fresnell and Arago explained the polarisation of light in terms of transverse light waves. However, none of these explanations was able to refute the particle hypothesis completely. Nevertheless, the wave versus particle theories differed in one fundamental aspect that could be tested. When light enters water it is refracted (Chapter 2). In terms of corpuscles, this implied a speeding up of the light in water relative to air. Thewave theory demanded that the light should move more slowly in water than in air. The experiments were complicated by the enormous speed of light, which was known to be about

Colour and the Optical Properties of Materials Richard J. D. Tilley 2011 John Wiley & Sons, Ltd Colour and the Optical Properties of Materials 2

gamma ultra- micro- radio rays X-rays violet infrared waves waves

10-14 10-12 10-10 10 8 10-6 10-4 10-2 1 102 wavelength (m)

vis ble

400 500 600 700 Wavelength / nm violet blue green yellow orange red Figure 1.1 The electromagnetic spectrum. Historically, different regions have been given different names. The boundaries between each region are not sharply deﬁned but grade into one another. The visible spectrum occupies only a small part of the total spectrum

3 108 ms 1, and it was not until April 1850 that Foucault first proved that light moved slower in water than in air, and seemingly killed the corpuscular theory then and there. Confirmation of the result by Fizeau a few months later removed all doubt. Over the years the wave theory became entrenched and was strengthened by the theoretical work of physicists such as Fresnel, who first explained interference and diffraction (Chapter 6) using wave theory. Polarisation (Chapter 4) is similarly explained on the assumption that light is a wave. The wave theory of light undoubtedly reached its peak when Maxwell developed his theory of electromagnetic radiation and showed that light was only a small part of an electromagnetic spectrum. Light was then imagined as an electromagnetic wave (Figure 1.1). Maxwell’s theory was confirmed experimentally by Hertz, whose experiments led directly to radio. The problem for the wave theory was that waves had to exist in something, and the ‘something’ was hard to pin down. It became called the luminiferous aether and had the remarkable properties of pervading all space, being of very small (or even zero) density and having extremely high rigidity. Attempts to measure the velocity of the Earth relative to the luminiferous aether, the so-called aether drift, by Michelson and Morley, before the end of the nineteenth century, proved negative. The difficulty was removed by Einstein’s theory of relativity, and for a time it appeared that a theory of light as electromagnetic waves would finally explain all optical phenomena. This proved a false hope, and the corpuscular theory of light was revived early in the twentieth century, principally by Einstein. Since 1895, it had been observed that when ultraviolet light was used to illuminate the surfaces of certain metals, negative particles, later identified as electrons, were emitted. The details of the experimental results were completely at odds with the wave theory. The electrons, called photoelectrons, were only observed if the frequency of the radiation exceeded a certain minimum value, which varied from one material to another. The kinetic energy of the photoelectrons was linearly proportional to the frequency of the illumination. The number of photoelectrons emitted increased as the intensity1 of the light increased, but their energy remained constant for any particular light source. Very dim illumination still produced small numbers of photoelectrons with the appropriate energy.

1 The imprecise expression ‘intensity’ has largely been replaced in the optical literature by well deﬁned terms such as irradiance (Appendix 1.1). The term intensity is retained here (in a qualitative way to designate the amount of light) because of the historical context. 3 Light and Colour

The explanation of this ‘photoelectric effect’ by Einstein in 1905 was based upon the idea that light behaved as small particles, now called photons. Each photon delivered the same amount of energy. If this was sufﬁciently large, then the electron could be ejected from the surface. The energy of each photon, E, was proportional to the frequency of the illumination, so that the photoelectron could be ejected when the frequency passed a certain threshold, but not before that point was reached. Thereafter, increasing the frequency of the illumination allowed the excess energy to be displayed as an increase in kinetic energy. The kinetic energy of the photoelectrons ejected from a metal under this hail of photons could then be written as:

1 mv2 ¼ Ef 2 where f is known as the work function of the metal and is simply the energy required to liberate the electron from the metal surface. The intensity of the light simply indicated the number of photons arriving at the surface, so that the number of photoelectrons emitted is a function of irradiance, but the energy of these electrons is a function of the frequency of the radiation. Einstein thus rescued the wave theory from the dilemma of the luminiferous aether and then seemingly wrecked the self-same theory via his explanation of the photoelectric effect. At present, all experiments show that light and its interaction with matter (i.e. atoms) is best described in terms of photons. At its simplest level, the statistical behaviour of a large number of photons is then represented very well by an electromagnetic wave. That is to say, photons are the components of a light beam, whilst waves are a mathematical description of a beam of light. In this book, explanations are given in terms of the simplest approach that is in accord with the observations. For large-scale phenomena, such as the operation of a magnifying glass, it is adequate to use the idea of a ray of light. When objects having dimensions of the order of hundreds of nanometres are encountered it is necessary to consider light to be awave.Atomic processes require a photon approach. It needs tobe stressed that these are not different fundamentally. All are contained within the theory of optics available today, generally described as quantum optics or quantum electrodynamics. No matter how it is described, light has no colour as such. Light simply leaves the generating source, possibly interacts with matter in the course of passage and then enters the eye. Colour, or more accurately the perception of colour, is the result of an eye brain combination that serves to discriminate between light of different wavelengths or energies. In the following chapters, the production of light and its interaction with matter is discussed from the point of view of colour that of the original light source, and how this is modiﬁed by interaction with matter to generate new colours.

1.2 Colour and Energy

Colour is generated by interactions of light and matter atoms and molecules, or, more strictly, the electrons associated with these. If light is considered as an electromagnetic wave, then the energy density of the wave, which is the energy per unit volume of the space through which the light wave travels, is given by:

2 E ¼ e0ðE0Þ where e0 is the vacuum permittivity and E0 is the amplitude of the electric component of the wave. Classical optics, the interaction of light with a transparent solid, in the main, is concerned with scattering of the light. This leads to the phenomena of reﬂection, refraction and so on. In these processes, colour is produced by interaction between various light waves, and energy exchange considerations hardly matter. These aspects of colour formation are covered in Chapters 2 6. Colour and the Optical Properties of Materials 4

When light is absorbed by or emitted from a material, say a gemstone such as ruby, energy changes are paramount. In this case, light is best regarded as a stream of photons; the energy of each photon being deﬁned as:

¼ n ¼ hc ð : Þ E h l 1 1 where n is the frequency of the equivalent light wave, l is the wavelength of the equivalent light wave, h is Planck’s constant and c is the velocity of light in vacuum. The absorption of light by isolated atoms or molecules involves a change in energy of the electrons surrounding the atomic nuclei. These occupy a series of atomic or molecular orbitals, each of which can be assigned a precise energy. The energies of the orbitals form a sort of ladder (with variable rung spacing) from low to high, each separated from the next by an energy gap. Electrons are fed into the orbitals from lowest to highest energy until all of the electrons have been allocated, leaving the extra outer electron orbitals empty. The total energy of all the electrons in the atom at modest temperatures is represented by an energy level called the ground state. The absorption of light will cause an electron to move from the low-energy ground state E0 to an empty orbital at a higher energy. The new energy situation is represented by an energy level at energy E1 (Figure 1.2a). (These energy levels and how they are enumerated will be described in detail in later chapters.) The relationship between the energy change DE and the frequency n or thewavelength l of the light absorbed is:

¼ D ¼ n ¼ hc ð : Þ E1 E0 E h l 1 2 where h is the Planck constant and c is the speed of light. When energy is lost from an isolated atom it moves from the excited state back to the ground state. The simplest case is when the species passes directly from E1 to E0 (Figure 1.2a), with an energy output given by:

¼ D ¼ n ¼ hc E1 E0 E h l identical to that of the absorbed radiation. However, the release of energy often takes place by more complex mechanisms that will be explored in later chapters. In both cases, if the frequency associated with the energy change DE lies in the band that is registered by the eye, then colour is perceived. When atoms unite to form a solid (or a liquid) the precise energies of the orbitals are broadened out into continuous bands of energy. The main energy landscape in a solid is the band structure which is the geometrical form of the energy bands throughout the matrix. In a solid, electrons are allocated to the energy bands, from the lowest energy up, until all have been allocated. The energy bands of highest energy are then empty, similar to the orbitals with highest energy in an atom. In the simplest depictions, the highest ﬁlled energy band (the conduction band) is separated from the lowest empty energy band (the valence band) by a constant band gap (Figure 1.2b). In real structures, the band architecture is more complex. Light absorption, emission and colour generation in a solid cannot be discussed without consideration of the role of the band structure. In this case, the energy difference DE which corresponds to colour registration might correspond to the promotion of an electron from a full conduction band to an empty valence band. However, impurities and defects can introduce further energy levels into the energy gap between the conduction and valence bands. In these cases, energy transitions between these levels or between them and the energy bands of the solid may then be of an appropriate energy to act as important sources of colour (Figure 1.2b). Examples of these instances are presented in Chapters 7 10. 5 Light and Colour

ΔE

(a)

conduction band

impurity energy levels

valence band

(b) Figure 1.2 Energy transitions leading to colour production, shown as arrows: (a) transitions between energy levels in isolated atoms or molecules; (b) transitions between impurity levels and energy bands in a solid. Note that each single energy level shown may actually be composed of several closely spaced energy levels in real systems. Eg is the magnitude of the energy gap between the valence and conduction band

1.3 Light Waves

In terms of thewave theory, light waves comprise a small segment of the electromagnetic spectrum (Figure 1.1). Any part of the electromagnetic spectrum is regarded as a wave of wavelength l with an electrical and magnetic component, each described by a vector and moving with a velocity, the ‘speed of light’, in a vacuum. The electric field vector2 E is perpendicular to the magnetic vector, described in terms of the magnetic induction B or the magnetic field H, and they are in phase, so that a peak in the electric field component coincides with a peak in the magnetic field component. Moreover, these vectors lie in a plane perpendicular to the direction in which the wave is moving, described by the velocity vector v, or the propagation vector or wave vector k. Thus, E and B both lie perpendicular to the direction of propagation, so that light is regarded as a transverse electromagnetic (TEM) wave. The wave is a progressive wave,atravelling wave or a propagating wave, all

2 Vectors are given in bold throughout this book. Colour and the Optical Properties of Materials 6 terms being used more or less interchangeably. The electric field vector, the magnetic field vector and the velocity vector can be represented by the three (right-handed) Cartesian axes (Figure 1.3a). As far as the topics in this book are concerned, the electric field E can usually be considered in isolation. In this case, a one-dimensional continuous electromagnetic wave moving in the þ x direction can be conveniently depicted by the equation:

Ey ¼E0 cos½ð2p=lÞðxvtÞ ð1:3Þ

v (k) x

H (B)

(a)

crest

ε0 distance x

trough

(b) Figure 1.3 Light waves. (a) Light can be thought of as a TEM wave. The electric (E) and magnetic (H or B) vectors lie perpendicular to each other and to the vector representing the direction of travel of the wave (v or k). The shaded planes represent the positions of peaks in the electric and magnetic fields. (b) Part of a light wave travelling along x. The curve represents the magnitude E of the electric field vector as a function of position. The distance between the crests or troughs is the wavelength l. Any point on the wave moves with a speed v.If the electric field vector remains in the plane of the paper, as drawn, the light is linearly polarised. If the orientation of the electric field with respect to the plane of the page varies at random so that the curve continually adopts differing angles with the plane of the paper, the light is unpolarised 7 Light and Colour

Here, Ey is the magnitude of the electric field vector at position x and time t and v is the wave speed 3 (or velocity ). The term E0 is the amplitude of the wave (the maximum value that the electric field vector takes) and is a constant. The speed v at which any point on the wave, say a peak or a trough, travels is called the phase speed or phase velocity. The velocity of an electromagnetic wave in vacuum, denoted by the speed of light c,is an important physical constant. Taking t as fixed gives a snapshot of the wave at a single instant (Figure 1.3b). The spatial period of the wave, which is the distance over which the wave subsequently repeats itself, is called the wavelength l. The peaks in the wave are referred to as crests and the valleys as troughs. The term in square brackets, ½ð2p=lÞðxvtÞ, i.e. the argument of the cosine function, is called the phase of the wave, represented by f. The phase of the wave is usually quoted in radians, in the form (mp/n), i.e. 3p/4. Clearly, the phase of the wave varies along its length and changes by 2p in one wavelength. The phase of a light wave cannot be determined. However, the phase difference between corresponding points on two different waves, say two equivalent crests, can be measured with considerable precision. Taking x as fixed will show that the magnitude of the electric field vector Ey will oscillate up and down between values of E0. The temporal period of the wave t, which is the time over which the wave subsequently repeats itself, is more usually encountered as the reciprocal 1/t and is equal tothe temporal frequencyn, which is the number of waves that pass a point per second. The speed of the wave v is related to the frequency n by:

v ¼ ln

(or in a vacuum by c ¼ ln). A beam of light is said to be monochromatic when it is comprised of a very narrow range of wavelengths and it is coherent when all of the waves which make up the beam are completely in phase; that is, the crests and troughs of all the waves are in step. The way in which the electric field vector is constrained describes the polarisation of the wave. If the electric field vector remains in one plane, then the light is said to be linearly (or plane) polarised. In general, the polarisation of the light wave must be considered when describing optical phenomena. Normal light, such as that from the sun, say, is not emitted in a continuous stream, but in short bursts lasting about 10 8 s. Within each burst all of the light waves are in phase and linearly polarised. However, both the phase and polarisation change from burst to burst in a random fashion, so that the phase and polarisation of each burst are unrelated to those in the preceding burst. This means that the phase and the polarisation of a light wave fluctuate continuously and at random within a fraction of a second. Normal light is thus described as being incoherent and unpolarised. Because of this, the interaction of daylight with objects can be interpreted (at least as a good approximation) without considering polarisation. Light from lasers (Section 1.9) is, by and large, coherent and polarised, and these aspects cannot usually be ignored.

1.4 Interference

One of the advantages of the wave description of light is that the interactions between two beams are easily explained. If two light waves occupy the same region of space at the same time then they add together, or interfere, to form a product wave. This idea, called the principle of superposition, was stated by Young some

3 Strictly speaking we are discussing wave speed, which is a scalar quantity. Velocityis a vector quantity. However, it makes things simpler to brush over this distinction in the present case. Colour and the Optical Properties of Materials 8 two centuries ago, in 1802. If two identical waves are exactly in step then they will add to produce a resultant wave with twice the amplitude (Figure 1.4a c) by the process of constructive interference. If the two waves are out of step, then the resultant amplitudewill be less, due to destructive interference. If the waves are sufﬁciently out of step that the crests of one correspond with the troughs of the other, then the resulting amplitude will be zero (Figure 1.4d f). Interference can occur between light waves with different relative frequencies, amplitudes and phases. However, for this to be observed the phase difference between the beams must remain constant. That is, the waves must be coherent. Many of the difﬁculties inherent in observing interference effects using normal light stem from the incoherent nature of thewave trains used, and efforts must be made to ensure that the incoherence does not destroy any visible interference patterns that may be generated. The use of laser light makes the observation of interference much simpler.

a0 (a)

a0 (b)

2a0

(c)

resultant

Figure 1.4 Interference of light waves. (a)–(c) The addition of two waves in phase, (a), (b), will produce a wave of twice the amplitude of the original wave, (c). (d)–(f) The addition of two waves out of phase by l/2, (d), (e), will produce a wave with zero amplitude, (f) 9 Light and Colour

a0 (d)

a0 (e)

(f) zero

resultant

Figure 1.4 (Continued)

The effects of interference can be assessed analytically using algebraic methods. An intuitive feeling for the phenomenon is best gained by adding waves represented by formulae such as Equation 1.3 using a computer and displaying the results graphically.

1.5 Light Waves and Colour

Our eyes can detect only a small part of the whole electromagnetic spectrum, called the visible spectrum (Figure 1.1). The amount of light that the eye records in any situation, which can loosely be called the brightness or intensity of the light, is not the amplitude of the wave but is the irradiance I, which is proportional to the square of the amplitude:

2 I ¼ KðE0Þ Colour and the Optical Properties of Materials 10

Table 1.1 The visible spectrum

Colour l/nm 10 14n/Hz 10 15o/rad s 1 1019 Energy/J Energy/eV Infrared 750 4.00 2.51 2.65 1.65 Deep red 700 4.28 2.69 2.84 1.77 Orange red 650 4.61 2.90 3.06 1.91 Orange 600 5.00 3.14 3.31 2.07 Yellow 580 5.17 3.25 3.43 2.14 Yellow green 550 5.45 3.42 3.61 2.25 Green 525 5.71 3.59 3.78 2.36 Blue green 500 6.00 3.77 3.98 2.48 Blue 450 6.66 4.19 4.42 2.75 Violet 400 7.50 4.71 4.97 3.10 Ultraviolet 350 8.57 5.38 5.68 3.54 where the value of the constant of proportionality K depends upon the properties of the medium containing the wave. (See Appendix A1.1 for information on units.) The extent of the visible spectrum is defined in terms of the wavelength or frequency of the light waves involved. Perception of the different wavelengths is called colour. The precise measurement of colour involves a determination of the energy present at each wavelength in the light using a spectrometer. The wavelength range that an eye can perceive varies from individual to individual. In general, it is assumed that the shortest wavelength of light that an average person can detect corresponds to the colour violet, with a wavelength near to 400 nm. Similarly, the longest wavelength of light registered by an average observer corresponds to the colour red, with a wavelength close to 700 nm. Between these two limits the other wavelengths of the spectrum are associated with the colour sequence from red to orange, green, blue, indigo and finally to violet (Figure 1.1 and Table 1.1). The divisions between these colours are, of course, artificial, and each colour blends into its neighbours. (Note that these colours are simply approximate labels for the wavelength. The perceived colour of an object is a function of a number of factors (Section 1.10).) It is known that the sensitivity of the eyes of animals is different than those of humans. Many insects, for example, can detect wavelengths shorter than humans but do not see so far into the red. Radiation with wavelengths shorter than violet falls in the ultraviolet region of the spectrum. Ultraviolet A (UVA) is closest to the violet region and is taken to have a wavelength range of 400 320 nm. This radiation is largely responsible for suntan. Ultraviolet B (UVB), with an approximate wavelength range of 320 280 nm, is more damaging and causes sunburn. Ultraviolet radiation with shorter wavelengths is called the far ultraviolet, (280 200 nm) and vacuum ultraviolet (below 200 nm). UVB and shorter wavelengths are able to damage biological cells severely, and excessive exposure leads to the occurrence of skin diseases. Radiation with wavelengths longer than red is referred to as infrared radiation. Although not visible, the longer wavelengths of infrared radiation, called thermal infrared, are detectable as the feeling of warmth on the skin.

1.6 Black-Body Radiation and Incandescence

There are many ways in which light can be generated, but the action normally tales place at an atomic level. Individual atoms (or molecules) lose energy, which is given out as radiation. These processes generally need to be discussed in terms of photons rather than waves. In this section, just one example is given, the generation of light by a hot body. This was the ﬁrst light-generating process to be understood at a fundamental level, and led directly to the photon concept as well as to an appreciation of our idea of the make-up of white light. 11 Light and Colour

Incandescence is the emission of light by a hot body. The sun and tungsten-lamp filaments provide commonplace examples, and both are regarded as producing (more or less) white light. The light characterising the upper part of a candle flame also arises from incandescence. In this case, small particles of carbon are heated to high temperatures in the flame and emit light which is perceived as yellow in colour. When light from an incandescent object is spread out according to wavelength by a prism (Chapter 2) the result is a continuous fan of colours following the sequence listed in Table 1.1 and called a continuous spectrum. The radiation emitted is both incoherent and unpolarised. Incandescence comes about in the following way. At absolute zero all atoms and molecules making up the solid are in the lowest possible energy state. As the temperature increases they absorb energy and are promoted to higher energies and, at the same time, atoms and molecules which have already absorbed energy lose some and they fall back to lower energies. (The energy levels involved in this process will be described in more detail in later chapters.) The radiation emitted in this way effectively extends over a continuous range of energies. For a solid a little above room temperature all the wavelengths of the emitted energy lie in the infrared; although the radiation is invisible, it is detectable as a sensation of warmth. At a temperature of about 700 C the shortest wavelengths emitted creep into the red end of the visible spectrum. The colour of the emitter is seen as red and the object is said to become red hot. At higher temperatures the wavelengths of the radiation given out extend increasingly into the visible region and the colour observed changes from red to orange and thence to yellow, as in the example of a candle flame, mentioned above. When the temperature of the emitting object reaches about 2500 C all visible wavelengths are present and the body is said to be white hot. The sun provides a perfect example, and the ‘colour’ white as applied to light is a combination of energies or wavelengths that spans the visible spectrum with the same composition as that of the radiation from the sun. These qualitative colour changes can be understood in terms of the radiation emitted by a black body. A black body is an idealized object which absorbs and emits all wavelengths perfectly. A reasonable approximation to a source of black-body radiation would be a small pinhole in the wall of a hot furnace. If the irradiance of the radiation issuing from the pinhole is measured as a function of wavelength, a characteristic curve is obtained called a black-body spectrum (Figure 1.5). The shape of the curve is dependent only upon the temperature of the body, and the maximum in the curve moves to shorter wavelengths as the temperature of the black body increases. The curve also mirrors the energy distribution inside the black body when in thermal equilibrium. The explanation of the form that this curve takes played a significant role in the physics of the twentieth century. Despite many attempts, the form of the black-body spectrum could not be explained by the classical wave theory of electromagnetic radiation. The successful theoretical description of this curve by Planck in 1901, now known as the Planck law of black-body radiation or Planck’s radiation law, signalled the start of the quantum theory. The equations describing the spectral radiance of all the radiation components within a black body at equilibrium at temperature T in the frequency range n to n þ dn or the wavelength range l to l þ dl are:

n3 2h 2 1 1 n ¼ ð : Þ L 2 units: W m sr Hz 1 4a c ½expðhn=kBTÞ1 or

2 2hc 3 1 l ¼ ð : Þ L 5 units: W m sr 1 4b l ½expðhc=lkBTÞ1

In these equations, h is a constant that is now called Planck’s constant, c is the speed of light, l is the wavelength, kB is Boltzmann’s constant and T (K) the temperature of the body. These equationsare often seen in Colour and the Optical Properties of Materials 12

5x1014 vis ble spectrum

4x1014 -3

3x1014 8000 K

2x1014 Spectral irradiance / W m

1x1014 4000 K 5000 K

500 1000 1500 2000 Wavelength / nm Figure 1.5 The radiation emitted from a black body as a function of wavelength. As the temperature of the body is increased, the maximum of the curve both increases and moves towards shorter wavelengths (higher energy). The spectrum emitted by the sun is similar to that for a black body at 6000 K and that from a red-hot object is similar to the curve for a black body at 1000 K

the form describing the spectral irradiance (if the energy falls upon a surface) or the spectral exitance (if the energy is observed after leaving a black body via a pinhole not large enough to disturb the thermal equilibrium within), In in the frequency range n to n þ dn or Il in the wavelength range l to l þ dl as a function of the wavelength l for a black body at a temperature T:

p n3 2 h 2 1 n ¼ ð : Þ I 2 units: W m Hz 1 5a c ½expðhn=kBTÞ1 or

p 2 2 hc 3 l ¼ ð : Þ I 5 units: W m 1 5b l ½expðhc=lkBTÞ1 or as the corresponding spectral energy density un in the frequency range n to n þ dn or ul in the range l to l þ dl as a function of the wavelength l for a black body at a temperature T:

p n3 8 h 3 1 n ¼ ð : Þ u 3 units: J m Hz 1 6a c ½expðhn=kBTÞ1 13 Light and Colour or

p 8 hc 4 l ¼ ð : Þ u 5 units: J m 1 6b l ½expðhc=lkBTÞ1

The revolutionary concept that Planck employed in the derivation of these equations to successfully reproduce the black-body curve was that the energy absorbed or given out by the atoms and molecules (the ‘oscillators’ in Planck’s time) in the black body could not take any value from a continuous spread of energies, but had to be delivered only in packets or quanta q0,2q0,3q0 and so on. The relationship between the energy of a quantum E and the frequency of the radiation n was given by what has since become one of the most famous equations of science:

E ¼ hn ð1:1Þ

The constant h, Planck’s constant, is one of the important fundamental physical constants. More recently, in the mid-twentieth century, it was realized that the cosmos was filled with some sort of background electromagnetic radiation. The peak of the radiation lies in the microwave part of the electromagnetic spectrum. Naturally, it is invisible to optical instruments and was first mapped using radio telescopes and latterly by satellites. The spectrum of this radiation fits that of a black body; and indeed, this radiation, called the cosmic microwave background radiation, is possibly the most accurately measured black-body radiation curve available. It is interpreted as lending strong support to the ‘Big Bang’ theory of the origin of the universe.

1.7 The Colour of Incandescent Objects

From the point of view of the colour of incandescent objects, one of the most important attributes of the emission curve is the variation in the position of the maximum as the temperature of the black body increases. (This was derived before the Planck radiation law and represents the ﬁnal success of classical electromagnetic theory.) The relationship, known as the Wien displacement law, is:

lmaxT ¼ constant where T (K) is the temperature of the body and the constant has a value of 0.002 898 m K. It can be derived from Equations 1.5a and 1.5b by differentiating with respect to l and setting the result equal to zero. The colour of an incandescent object is then controlled by the maximum of the black body curve (or an approximation to it), as mentioned below. The second factor of importance is the spread of the spectrum. A cool body will be perceived as initially showing a colour when the peak of the curve is close enough to the visible range that some radiant energy creeps into the low-energy (red) end of the spectrum. As the temperature of the incandescent object increases, the peak moves to higher energies, following the displacement law, and the spread moves further across the visible spectrum, resulting in the colour sequence of dull red, red hot to white hot to blue white. The colour of an incandescent object is described by its colour temperature if the spectrum resembles that of a black body closely. Most solids behave like black bodies over some range of temperature and wavelength, and stars are a close approximation over the whole of the wavelength range. If the match is approximate, the term used is correlated colour temperature and this expression is used for light sources that are not incandescent, Colour and the Optical Properties of Materials 14

Table 1.2 Colour temperature of incandescent sources

Light source Correlated colour temperature/K Mean noon sunlight 5 400 Electronic flash 7 000 Blue flash bulb 6 000 Tungsten filament photographic lamps 3 400 Tubular triphosphor fluorescent lamp, 36 W 3 000 Household tungsten filament light bulb, 100 W 2 850 Standard candle 1 930

Table 1.3 Effective star temperatures

Star colour and example Effective temperature/K Blue white, Bellatrix 25 000 White, Sirius 11 000 Yellow white, Sirius Solar 7 500 Solar, the Sun 6 000 Orange yellow, Arcturus 4 200 Orange, Antares 3 000 Deep orange red, m Cephei 2 600

such as fluorescent lighting (Table 1.2). Colour photographs taken on film designed to be used in daylight (colour temperature of about 5 400 K) will show incorrect tones when used to photograph objects illuminated with tungsten lights (colour temperature of about 3 400 K) or fluorescent lights (colour temperature of about 3 000 K) unless correcting filters are used. The most important incandescent object for us is the sun, which is the ultimate source of energy on Earth. The solar spectrum has a form quite similar to a black-body curve corresponding to a solar temperature of about 5 780 C (about 6 000 K), which has a maximum near 480 nm. The form of the spectrum when it reaches the surface of the Earth is a function of a number of variables, including the elevation of the sun, the amount of scattering material in the atmosphere and so on. Light is perceived as white if it has a make-up like that of the solar spectrum from an overhead sun on a clear day. Stars which are cooler than the sun give a redder colour, whilst those which are hotter are perceived as whiter. The effective temperature of a star is the temperature calculated as if it were a black body radiating with the same energy over the same wavelength ranges (Table 1.3). The effective temperature is generally a good approximation to the surface temperature of a star. The hottest visible stars are the Bellatrix type, with blue white colour and an effective temperature of approximately 25 000 K, whilst the reddest naked-eye star is m-Cephei, the Garnet Star, with a temperature of approximately 2 600 K.

1.8 Photons

The quantization of radiation proposed by Planck in the derivation of the radiation law was not seized upon instantly. After a lapse of some years it was exploited by Einstein in his explanation of the photoelectric effect 15 Light and Colour in 1905 (Section 1.1). He proposed that the quantization of radiation contained in Planck’s formula for black- body absorption and emission of energy, i.e.:

E ¼ hn where nwas the frequency of the radiation and h is Planck’s constant, could be applied to the radiation itself, not just to the energy exchangewith atoms or molecules. That is to say, light was to be regarded not as awave butas a hail of bullet-like objects (which are now called photons), each of which had an energy hn. Each photon delivered the same amount of energy. If this was sufﬁciently large then the electron could be ejected from the surface. The energy of each photon was proportional to the frequency of the illumination, so that when the frequency passed a certain threshold, the photoelectron could be ejected, but not before that point was reached. Thereafter, increasing the frequency of the illumination allowed the excess energy to be displayed as an increase in kinetic energy. The kinetic energy of the photoelectrons ejected from a metal under this hail of photons could then be written as:

1 mv2 ¼ hvf 2 where f is known as the work function of the metal and is the energy required to liberate the electron from the metal surface. The irradiance of the light indicated the number of photons arriving at the surface, so that the number of photoelectrons emitted is a function of irradiance, but the energy of these electrons is a function of the frequency of the radiation. A description of light in terms of photons is mandatory when dealing with events at an atomic scale. The energy E of a photon is given by Equation 1.1:

¼ n ¼ hc ð : Þ E h l 1 1 where n is the frequency of the equivalent light wave, l is the wavelength of the equivalent light wave, h is Planck’s constant and c is the velocity of light in vacuum. This conjunction of the particle and wave descriptions, called wave particle duality, is evident in the fact that n is the frequency and l is the wavelength of the wave-like properties associated with the photon. In fact, all particles exhibit wave-like properties. The momentum p of a particle (such as an electron, say), is given by:

= ðE2m2c4Þ1 2 p ¼ c where E is the energy, m the particle mass and c the speed of light. For a photon, m ¼ 0, so that:

E p ¼ c

The wavelength of a particle is:

l ¼ h ¼ hc = p ðE2m2c4Þ1 2 Colour and the Optical Properties of Materials 16

For a photon, m ¼ 0, so that:

l ¼ hc E The velocity of a particle is:

= pc2 m2c4 1 2 v ¼ ¼ c 1 E E2

For a photon, m ¼ 0, so that:

v ¼ c

(For particles such as electrons, m is not zero.) For many purposes the wave and particle aspects of light can be used interchangeably, as dictated by experiment.Thewaveaspectoflightexpressesthefactthatthephotonsdonotobeydeterministiclawsofmotion, but laws of probability. The waves associated with light photons are a way of describing these probabilities.

1.9 Lamps and Lasers

1.9.1 Lamps

Until the end of the nineteenth century artificial illumination was via incandescence either firelight, candles, oil lamps or gas light. At the end of this period, new light sources began to be invented in parallel with the generation and availability of electricity. In 1897 Nernst invented the ‘glower’. This lamp consisted of a bar of electrically conducting ceramic made from a mixture of lanthanide oxides that became incandescent under the action of an electric current. Although Nernst glowers were widely used and were more efficient than the competing incandescent carbon-filament electric light bulbs developed by Edison, they fell into disuse following the successful introduction of tungsten-filament lamps after the invention of the Coolidge process for the production of ductile tungsten wires for the fabrication of lamp filaments. Throughout the twentieth century, tungsten-filament lamps dominated the lighting market. Although incandescence was the most widespread source of artificial light, other lighting was well known. Neon signs (Chapter 7) and various forms of luminescence (Chapter 9) were used in specialist light-generating ways, such as, in the case of neon signs, for advertising. These latter mechanisms relied directly upon atomic transitions in a way that was obscured in the complex incandescence reactions. In addition, instead of generating a continuous ‘white light’ spectrum, these new light sources tended to give out coloured light, the wavelengths produced depending upon the actual atoms emitting the photons. All of these light sources, however, were similar to each other in one way the light emitted was incoherent and usually unpolarised. Towards the middle of the twentieth century, advances in communications technologies reinforced the utility of using light directly to carry signals. This necessitated the use of coherent radiation. Initially the push came from radio, as radio waves are normally emitted as a coherent wave train, not as incoherent waves. Thewavelength of thewaves used for carrying signals continually decreased via long waves, medium waves and short waves. At the same time, the engineering skills required to encode greater and greater information on these waves increased to an amazing extent, making television and stereo broadcasting a norm. Unfortunately, the production of coherent radiation seemed to be stuck somewhere in the microwave region. The idea of using lower wavelengths, though, especially optical wavelengths, was enormously attractive, and a 17 Light and Colour great deal of effort was invested into breaking into this wavelength range. Success came in the 1960s, with the invention of the laser. Lasers are a completely new sort of lamp compared with those already described. The word laser is an acronym for the expression Light Amplification by Stimulated Emission of Radiation. The first laser to be made was the ruby laser, and the first laser light emitted was on 15 May 1960. Since then a vast number of lasers have been produced, including solid-state lasers, gas lasers, semiconductor diode lasers and dye lasers. From an exotic beginning lasers have become ubiquitous in modern life, being used as pointers, at check-outs in supermarkets, in surveying and measurement, in micromachining, microsurgery and so on. Here, the general principles of laser action will be outlined. Examples which illustrate particular facets of laser light generation will be discussed throughout the text.

1.9.2 Emission and absorption of radiation

When a photon of energy hn is absorbed by an atom or molecule it passes from the normally occupied lower energy state, often called the ground state, to an upper or excited state, as described above. The transition will take place if the frequency of the photon n, is given exactly by:

¼ D ¼ n ¼ hc ð : Þ E1 E0 E h l 1 2 where E0 is the energy of the ground state, E1 is the energy of the excited state and h is Planck’s constant. If the atom is in the excited state E1 and makes a transition to the ground state E0, energy will be emitted with the same frequency, given by the same equation. In this description the actual emission mechanism is ignored. In 1917 Einstein suggested that there should be two possible types of emission process (Figure 1.6): 1. An atom in an excited state can randomly change to the ground state, by a process called spontaneous emission. 2. A photon having an energy equal to the energy difference between the two levels (i.e. E1 E0) can interact with the atom in the excited state, causing it to fall to the lower state and emit a photon at the same time, a process called stimulated emission. The light emission from ‘ordinary’, i.e. non-laser, sources is the result of spontaneous emission. Lasers are concerned with stimulated emission. In spontaneous emission, the light photons all have the same frequency but possess random phases and polarisation so that the light is incoherent. In stimulated emission the photon produced has the same frequency, phase and polarisation, as the one which caused the emission so that the light is coherent. It is these important features of stimulated emission on which the special properties of laser light depend.

1.9.3 Energy-level populations

Under conditions of thermal equilibrium the relative populations of a series of energy levels will begivenby the Boltzmann law, which for two energy levels can be written as: N ðE E Þ 1 ¼ exp 1 0 N0 kBT where kB is Boltzmann’s constant, T is the absolute temperature, E1 and E0 are the energies of the excited state and the ground state respectively and N1 and N0 are the numbers of atoms (the populations) in each of these energy levels. For ordinary atoms, in a gas, liquid or solid at ordinary temperatures, the fraction N1/N0 will be Colour and the Optical Properties of Materials 18

E1 E1

hν

E0 E0 light absorbed spontaneous emission (a) (b)

E0 ν h hν hν

E0 stimulated emission (c) Figure 1.6 Light absorption and emission. (a) Light absorption occurs when a photon excites an atom (or molecule) from the ground state E0 to an excited state E1. (b) During spontaneous emission, the atoms lose energy and release photons at random. (c) During stimulated emission, an atom in an excited state is triggered to lose energy by interaction with a photon of energy (E1 E0 ) negligible for energy levels which are sufficiently separated to give rise to visible light. Atoms can be assumed to be in the ground state as far as visible light emission is concerned. When a photon of the appropriate energy interacts with an atom in the ground state it will be absorbed and shortly afterwards re-released by spontaneous emission (Figure 1.7a). This will be repeated at each atom in the ground state. There will be no amplification and we may well see a net absorption of energy. To obtain laser amplification one needs to ensure that stimulated emission is the dominant process occurring. This means that there are more atoms in the excited state of energy E1 than in the ground state E0. In this instance, a photon interacting with an excited atom can cause energy to be released by stimulated emission and two photons emerge. If most atoms are in the excited state then amplification may occur (Figure 1.7b). The situation in which more atoms are in the excited state than in the ground state is called a population inversion. From the Boltzmann equation it is obvious that an increase in temperature cannot achieve this objective. Even an infinite temperature will only result in equal numbers of atoms in E0 and E1. To obtain a population inversion, therefore, a nonequilibrium state must be achieved. The crux of laser action is how to create such a nonequilibrium situation in a material and then exploit it to produce the desired amplification. Examples of practical ways in which this is achieved are given later in the text (i.e. see Chapters 7 and 10).

1.9.4 Rates of absorption and emission

In the previous section it was implicitly implied that the rate of spontaneous emission was fast. This aspect must be looked at in more detail to obtain a better understanding of laser action. When equilibrium between absorption and emission holds, the rate of depopulation of an upper level (dN1/dt) by spontaneous emission 19 Light and Colour

hν hν

(a)

E1 hν

(b) Figure 1.7 Amplification. (a) When most atoms are in the ground state the absorption of a photon and the subsequent spontaneous re-emission will not lead to amplification. (b) When most atoms are in the excited state, stimulated emission can lead to amplification will be given by a first-order rate law:

dN 1 ¼ A N dt 10 1 where the negative sign denotes that the number N1 of atoms in the upper state E1 (per cubic metre, say) is decreasing with time. The rate is proportional to the number of atoms N1 in the state. The rate constant, denoted here as A10, is called the Einstein coefﬁcient for spontaneous emission, where the sufﬁx ‘10’ means that we are considering a transition from the excited state E1 to the ground state E0. The number of downward transitions due to spontaneous emission, per second, will be given by:

A10N1 Similar rate laws can be written for the cases of stimulated emission and for absorption, but in this case the rates are proportional to the numbers of atoms in the relevant state and, in addition, the number of photons present. The reactions can be taken to be ﬁrst order with respect to both of these quantities. The rate at which atoms in state E0 are excited to state E1 is then given by:

dN 0 ¼ B rðn ÞN dt 01 01 0 where N0 is the number of atoms in state E0 (per cubic metre, say), r(n01) is the radiation density responsible for absorption, which is the number of quanta per cubic metre incident per second at the correct excitation frequency n01, and B01 is the Einstein coefﬁcient for absorption of radiation. Similarly, the rate of depopulation Colour and the Optical Properties of Materials 20

of state E1 by stimulated emission is given by:

dN 1 ¼ B rðn ÞN dt 10 10 1 where N1 is the number of atoms in state E1 (per cubic metre), r(n10) is the radiation density responsible for depopulation, which is the number of quanta per cubic metre incident per second at the correct frequency n10, and B10 is the Einstein coefﬁcient for stimulated emission of radiation. Now, the correct frequency for excitation will be the same as that for depopulation, so that n10 ¼ n01, which we can simply write as n, and the radiation density will be the same in each case, so that we can write:

rðn10Þ¼rðn01Þ¼rðnÞ

The number of stimulated downward transitions per second will be given by:

N1B10rðnÞ while the total number of upward transitions in the same time will be given by:

N0B01rðnÞ

At equilibrium, the total number of transitions in each direction must be equal; hence:

N0B01rðnÞ¼N1A10 þ N1B10rðnÞ so

rðnÞ¼ N1A10 N0B01N1B10

In addition, at equilibrium the Boltzmann distribution applies; thus: N hn 1 ¼ exp N0 kBT and by making this substitution we have:

rðnÞ¼ A10 expðhn=kBTÞB01B10

This expression represents the radiation density at frequency n. At thermal equilibrium, this should be identical to Planck’s equation, Equation 1.6a:

p n3 rðnÞ¼ 8 h 3 c ½expðhn=kBTÞ1 21 Light and Colour which leads to the conclusion that:

B01 ¼ B10 ¼ B and:

p n3 A10 ¼ 8 h B c3

The ratio of the rate of spontaneous emission to stimulated emission under conditions of thermal equilibrium is given by: A hn R ¼ 10 ¼ exp 1 rðnÞB kBT

This is an extremely interesting result. At 300 K, at visible wavelengths, R 1. This shows that, for light, stimulated emission will be negligible compared with spontaneous emission and reinforces the idea that it will be impossible to make a laser under equilibrium conditions. On the other hand, if the wavelength increases beyond the infrared into the microwave and radio-wave regions of the electromagnetic spectrum, R becomes much less than unity and all emission will be stimulated. Hence, radio waves and microwaves arise almost entirely from stimulated emission and are always coherent. This is one of the main reasons that communications in the early part of the twentieth century used radio waves. Perhaps because of this equation, and the towering reputation of Einstein, it seems that for the first part of the twentieth century it was felt that lasers were not feasible. In the middle of the century, scientists started to explore stimulated emission at microwave frequencies, developing the maser. This soon led to the first lasers, the ruby laser and then the He Ne gas laser, produced in 1960 with these early devices often being called optical masers. Once the way to overcome the production of laser light was understood, laser development became prolific. Later sections show how the equilibrium problem has been bypassed and how the difficulty of achieving stimulated emission at optical wavelengths has been overcome.

1.9.5 Cavity modes

Supposing that a population inversion is obtained between energy levels that would give rise to visible light, it is still necessary to design the equipment so that amplification of the signal takes place. The losses from the laser must be less than the total emission for amplification to be achieved. Losses in oscillating systems are often defined in terms of a quality factor Q, a term borrowed from radio technology. In effect, a high value of Q is needed to ensure amplification. One of the most important of these design features is the shape of the cavity that the laser medium occupies. Suppose that this is simply a crystal rod. The population is an unstable state and after a short time some spontaneous emission will occur from E1. Naturally, these photons will rapidly leave the crystal rod; and although in so doing a few other atoms might lose energy via stimulated emission, no amplification will occur. It is necessary to prevent the photons from leaving the crystal in order to increase the chances of stimulated emission occurring. The simplest way to achieve this is to coat the ends of the crystal rod with a highly reflecting mirror. In this case the photons are reflected to and fro, causing stimulated emission from the other populated E1 levels. Once started, the stimulated emission rapidly depopulates these levels in an avalanche. In order to permit some light to emerge, one of the mirrors is not perfect and allows a small Colour and the Optical Properties of Materials 22 amount of light to pass. There will then be a burst light emerging from the cavity which is not only coherent but also shows amplification. Thus, the simplest cavity geometry is simply cylindrical with one end fitted with a completely reflecting mirror and the other with an almost perfect mirror, appropriate to the wavelength of the light generated by the stimulated emission. There are several consequences of this simple geometry which are easiest to explain if the light trapped in the cavity is regarded as a wave. Taking the cavity as a rod with reflecting end faces, it is clear that initially all photons will be emitted at random, but only those that are emitted more or less parallel to the long axis of the cavity will bounce to an fro and so cause the stimulated emission avalanche. In terms of wave optics, the photons form a series of standing waves in the cavity, which is described as resonance. The standing waves form only if there is a node at each reflecting surface. The allowed waves are called longitudinal cavity modes and are given by the condition that a complete number of half wavelengths must fit into the length l of the cavity, i.e.:

l 2l m ¼ ¼ c l=2 l where mc is an integer, l is the cavity length and l is the wavelength of the mode. The frequency of a mode is given by: v n ¼ mc m 2l

where v is thevelocity of the light waves in the cavity, given by nml, and nm is the frequency of the mode mc. The separation of the modes is given by: v n n ¼ Dn ¼ m m 1 2l

The velocity of light in the cavity is given by:

v ¼ c n where c is the velocity of light in a vacuum and n is the refractive index of the cavity medium (Chapter 2), so that:

n n ¼ Dn ¼ c m m 1 2ln

How does this work out in practice? The emission from the upper to the lower energy level has been written as a single energy with a negligible width. In the case of real materials, atoms and molecules are in continuous motion, vibration in solids, translation in gases, and the sharp energy levels idealized in Figures 1.6 and 1.7 give rise to a spread of energies (or of frequencies or wavelengths) called the transition bandwidth (Figure 1.8a). Only that part of this output that fulﬁls the longitudinal mode criterion will be allowed to grow. The output from the cavity will then be composed of a set of modes (Figure 1.8b). These modes will depend upon the shape of the initial emission pulse and the overall power of the excitation process. 23 Light and Colour

(a)

emission Irradiance

Frequency

(b) / 2ln

cavity modes Irradiance

Frequency

Figure 1.8 (a) The emission from an excited state E1 to the ground state E0 is not sharp, but consists of a range of frequencies dependent upon temperature and other factors. (b) In a laser cavity, only certain frequencies, the cavity modes, are allowed to propagate

By extension, it is apparent that, in general, there will be transverse modes as well as longitudinal modes in the laser emission. These must be taken into account when the optics of the laser beam are considered. Laser cavity design is, therefore, of considerable importance in practice.

1.10 Vision

As stated earlier, light has no colour as such. Light radiation leaves the source, possibly interacts with matter in the course of passage and then enters the eye. Light is perceived by the eye brain combination, and colour is a description of this perception. The colour that the observer is conscious of is thus a combination of many factors, including the energy spread of the source light, the addition or subtraction of energy during any interactions with other materials and the sensitivity of the eye. For example, the blue sky contains all the colours of the spectrum, as can be demonstrated by passing this light through a prism (Chapter 2). Blue is the colour attributed to the sky when all the factors mentioned above are taken into account. Colour and the Optical Properties of Materials 24

The physiological response of the eye brain combination arises when light waves fall upon the light- sensitive retina, which makes up the inner surface of the eye. In 1876 Boll reported that the red purple pigment found in this part of the eyes of animals bleached in the presence of light to a colourless form. The change was found to be reversible, and in the dark the purple colour was regenerated. This important photochromic reaction is the source of vision. The compound involved became known as ‘visual purple’ and is now called rhodopsin. Vision in humans and other animals involves a complex set of reactions which take place in two types of photoreceptor cells located in the retina of the eye:rods and cones. There are about 108 rods and 4 106 cones in an eye. In humans the rod cells, of about 0.002 mm diameter, are about four times as sensitive as cones and are responsible for vision at low light intensities. Although they detect light all across the visible, the peak sensitivity is at 500 nm. The light not absorbed, red and blue/violet, gives rise to the purple colour of the membrane. The rod cells are not sensitive to colour and give rise to a monochrome image. Moreover, they saturate in high light levels, making them unresponsive under these conditions. The cone cells, approximately 0.006 mm diameter, are sensitive to bright light and form the daylight colour detection system. They exist in three varieties with peak sensitivities in three different regions of the visible: L cones, most sensitive to red, l(peak) 560 nm, M cones, most sensitive to green, l(peak) 530 nm, and S cones, most sensitive to blue l(peak) 420 nm (Figure 1.9a). The human eye is optimally sensitive to green light and is noticeably less sensitive to red and especially blue light (Figure 1.9b). The sensitivity of the eye to colour depends not only upon the amount of light, but also upon which area of the retina is being stimulated. The most sensitive region, called the fovea, is almost directly behind the lens of the eye and predominantly contains cone cells. The maximum sensitivity of a normal eye to bright white light focused on the fovea, which is the sum of the contributions of the cone and rod cells, is for a wavelength close to 555 nm (Figure 1.9c). Colour blindness results from a fault or deﬁciency in one or more varieties of the cone cells or in the way in which these cells communicate with the brain.4 Human vision is said to be trichromatic. There is considerable variation across the human population in the sensitivity ranges of the cone cells, giving rise to avariation in colour vision. Trichromaticity is common among primates, but most nonprimate animals can only detect two colours and are referred to as dichromats. However, some birds, ﬁsh and reptiles have four different cone cell receptors and can detect ultraviolet light with l(peak) as low as 360 380 nm in addition to three ‘normal’ colours. When light photons impinge on both rod and cone cells they are absorbed by stacks of photoreceptor molecules which are bleached in the process. This sends a nerve impulse to the brain. The system is remarkably sensitive and there is considerable evidence to suggest that in the rod cells just one photon is enough to stimulate the nerve. The light-absorbing pigments consist of a protein, an opsin, bound to a light-absorbing molecule, retinal. The receptor in the rod cells is called rhodopsin, while those in the cone cells are called cone opsins. The opsin part of the receptor, consisting of 364 amino acid residues in humans, is arranged in the form of seven helices, which penetrate the cell wall and enclose the retinal, which is bound to the amino acid lysine 296 (Figure 1.10). The opsin proteins differ from one cone cell to another and from rhodopsin in the rods, and it is these differences that confer the differing sensitivities to the receptors. However, the differences are rather small. For example, the amino acid sequences in the green (M) and red (L) cone receptors in humans differ in only three of the amino acid residues in 364.

4 The existence of colour blindness itself was ﬁrst recorded as such by John Dalton, who realized that his own perception of colours was different than the majority of his friends (but the same as his brother’s), and for many years the condition was known as Daltonism. A more recent study of his careful observations suggests that he was unable to distinguish the colour red. It is of interest to learn that Dalton himself felt that he possessed some visual advantages over his friends because of the nature of the abnormal sensitivity of his eyesight. He did not ﬁnd that he was at a disadvantage at all. (Also see Figure 1.15.) 25 Light and Colour

(a)

420 nm 534 nm 564 nm Absorbance (normalised)

400 450 500 550 600 650 700 Wavelength / nm

(b) green

red Relative sensitivity

blue

400 450 500 550 600 650 700 Wavelength / nm

Spectral luminous efficiency 0.2 0.1

400 450 500 550 600 650 700 Wavelength / nm Figure 1.9 The sensitivity of the eye to light, schematic. (a) Sensitivity of the cone cells in a normal eye to light as a function of the wavelength. (b) Visual sensitivity of a normal eye to red, green and blue light. (c) Overall visual sensitivity of a normal eye to light; the photopic spectral luminous efﬁciency function. The maximum sensitivity is for a wavelength close to 555 nm Colour and the Optical Properties of Materials 26

(a)

(b) opsin helices retinal

Figure 1.10 (a) The schematic structure of an opsin protein in the cell wall of a photorectetor. (b) The opsin protein molecule is in the form of seven helices arranged to enclose a retinal molecule. [(a) is adapted from http:// en.wikipedia.org/wiki/Rhodopsin]

In humans, retinal is derived from the compound b-carotene (Section 8.5), an orange pigment found in carrots. This is transformed into vitamin A in the liver, which then forms retinal. Thevisual pigments in animals then consist of retinal plus opsin. (There are two forms of vitamin A: A1, which gives retinal1 (11-cis-retinal, the aldehyde of vitamin A1), and A2, which gives retinal2 (3-dehydro-retinal). Retinal1 is used by all mammals and birds and will just be referred to as retinal in what follows.) The framework of the processes triggering vision is well established. It is described here with respect to rod cells, which have been studied in most detail. The chromophore (light absorbing part) of rhodopsin is the cis- form of the molecule retinal, 11-cis-retinal (Figure 1.11a). This cis-retinal molecule is bound to the opsin via the amino acid lysine, to form rhodopsin (Figure 1.11b). The cis-retinal by itself is not coloured and has an absorption maximum between 370 and 380 nm. However, when joined to the opsin the absorption maximum moves to about 500 nm. Molecules which can cause the deepening of the colour of a chromophore are called bathochromes and the resultant movement of the absorption maximum is referred to as a bathochromic shift. The bathochromic shift comes about because of the particular conformation of the cis-retinal molecule in conjunction with the protein. The bonding and slight differences in the various forms of the opsin molecules produce different bathochromic shifts and, hence, make the cones sensitive to the different wavelengths of red, green and blue light. The molecular mechanism leading to the nerve impulse hinges on the fact that retinal can exist in two isomeric forms, the cis-form already described and a trans-form, called all-trans-retinal. Under the inﬂuence of a photon the cis-retinal molecule changes to all-trans-retinal rhodopsin (Figure 1.11c). Absorption of light by rhodopsin drives the molecule through several intermediates to the bleached state, which can consist of a number of different molecules (metarhodopsin I, metarhodopsin II and so on), depending upon the conditions experienced. Thereafter the reaction reverses, again passing through a number of intermediates, so that the trans-retinal readopts the cis-conformation and reforms rhodopsin (Figure 1.11d). Another photon can trigger the cycle again. Each cycle takes only a fraction of a second and can repeat indeﬁnitely in normal light conditions so as to send a stream of nerve impulses to the brain. These impulses end when the light is extinguished and all molecules revert to rhodopsin. It is worth commenting on the enormous complexity of vision. The description of the cycle occurring in rod cells and presumed to occur in the cone cells described above is only true at moderate light intensities. At lower light intensities the trans-retinal molecule in rod cells is released completely from the opsin. Two processes then operate, dependent upon the weakness of the light signal. At the ‘higher’ of these lower intensities the trans-retinal is transformed back to the cis-conformation by the action of enzymes in the eye itself, whereupon 27 Light and Colour

H3C CH3 CH3

7 9 11 8 10 12

13 11-cis-retinal (a) CH 14 3 H3C

15 CHO

HC3 CH3 CH3

7 9 11 8 10 12

13 rhodopsin 14 CH3 H3C

15 + (b) N opsin H

light photon

HC3 CH3 CH3 CH3

7 9 11 13 15 8 10 12 14 N+ opsin H

CH3 all-trans-retinal rhodopsin (c)

light photon

all-trans rhodopsin rhodopsin

intermediates intermediates

metarhodopsins (d) Figure 1.11 The structures of (a) 11-cis-retinal, (b) rhodopsin and (c) all-trans-retinal rhodopsin, produced by the action of light on (b); (d) cycle of chemical changes producing vision. In normal illumination this process is repeated many times a second. Each cycle results in the transmission of a signal along the optic nerve to the brain Colour and the Optical Properties of Materials 28 the molecule is reattached to the opsin. At the lowest light intensities, the trans-molecules actually leave the eye completely, enter the bloodstream and are reprocessed to the cis-form in the liver, an occurrence which contributes to the length of time that it takes to become fully ‘dark adapted’. Rhodopsin has another role to play in the broader picture of life. It has been found that some purple halobacteria, bacteria which inhabit very salty environments, are coloured purple by a version of rhodopsin called bacteriorhodopsin. This consists of 247 amino acid residues, arranged in seven helices, with the photoactive retinal attached to lysine 216. It is, however, not used for vision, but in an analogous fashion to chlorophyll in plants. Absorption of light by chlorophyll initiates a chain of electron transfer reactions which eventually provide the energy for plant growth. In the purple halobacteria, the rhodopsin converts sunlight into energy for the metabolism of the bacterium. In essence it appears that the cis trans change acts as a proton pump, and the resulting electrochemical potential created initiates the energy building steps.

1.11 Colour Perception

Recognition of colour is a function not only of the physical make-up of the light falling on the eye and physiological factors, but also of psychological biases. The ‘colour’ of an object in this sense is changed by factors such as surface roughness or texture. Subsurface scattering, which returns some incident light on a body to the exterior, is of importance in the appearance of skin, cosmetics and paint. Because of this interplay it is possible to distinguish a hard red plastic surface from a red velvet surface even though in terms of physics the colours of both may be identical, originating in the same dye or pigment. It is clear that when describing the appearance of an object incolour terms it is necessary to consider specular (mirror-like) reflection, diffuse (non- mirror-like) reflection and subsurface scattering, as well as the make-up of the light which is reflected or scattered. Moreover, human eyes vary in colour-interpreting ability. It appears that an average person can distinguish more than a million different colours. All of these aspects are implied when the colour of an object or a light source is mentioned in a colloquial way. Because of this, colour is difficult to quantify. Despite the complexity inherent in the concept of colour and its perception, it has been found that all colours can be precisely specified by three parameters. Colours can then conveniently be represented by points in a three-dimensional coordinate system. There are many diagrammatic ways of representing the three attributes, and these are called colour spaces. The way in which the coordinates of any colour in the colour space are derived is called a colour model. There are many colour models, of which only three will be described briefly in this book. (More information can be found in Section 1.17.) One widely used colour model takes as initial parameters the three attributes hue, saturation and brightness to give the HSB model. These characteristics are generally taken to be: 1. Hue, which corresponds to thewavelength or frequency of the radiation. The hue is given a colour name such as red or yellow. 2. Saturation or chroma, which corresponds to the amount of white light mixed in with the hue and allows pale ‘washed out’ colours to be described. 3. Brightness, lightness, luminance,orvalue, which describes the intensity of the colour, the number of photons reaching the eye. This model is also given the acronyms HVC (hue, chroma, value), HSL (hue, saturation, luminance), HIS (hue, intensity, saturation) and HCL (hue, chroma, luminance). One way of building a colour space in terms of this colour model is to arrange the hue around the periphery of a disc with the degree of saturation of the colour represented by the distance from the centre of the disc along the radius. Brightness is defined by an axis perpendicular to the centre of the disc (Figure 1.12a). This arrangement has been quantified in constructions such as the Munsell colour cylinder or Munsell colour solid (Figure 1.12b). 29 Light and Colour

brightness / white

red

purple yellow

saturation green

blue hue (a) brightness / black

white

(b) black Figure 1.12 The representation of colours on a cylindrical colour space in the HSB colour model. (a) The hue is given by a point on the circumference of a planar disc, the saturation by the distance along the radius from the centre of the disc and the lightness by the vertical axis of the system. (b) The solid representation of the colours forms a colour cylinder, the best known of these being the Munsell colour cylinder [adapted from the Epson Online Printer Guide]

1.12 Additive Coloration

Additive colour mixing occurs when two or more beams of differently coloured light combine (i.e. overlap on a perfectly white surface, or arrive at the eye simultaneously). Colours on television screens are produced by additive coloration, as the screen is composed of small dots of three different phosphors each of which shines with one of three primary colours when activated. Additive coloration is also used in the painting technique known as pointillism. In this method of painting, the image is built up by placing small dots of relatively saturated colour onto the canvas, making sure that they do not overlap. When viewed from a distance of a few metres such pictures appear bright and dynamic. The colour patterns on the wings of many butterflies and moths are produced in a similar way. The wings are tiled with a fine mosaic of scales, each of which reflects only one colour. The colour perceived by the eye is an Colour and the Optical Properties of Materials 30 additive colour arising from the numerous closely spaced scales. The range of colours which can be produced by rather a few basic pigments is remarkable. For example, some perceived purples arise from mixtures of black, white and red scales, while some greens arise from mixtures of yellow and black scales. It has been found that the majority of additive colours can be produced by mixing just three additive primary colours, red, green and blue. (Strictly speaking, any fairly monochromatic light near to these colours will suffice). Moreover, mixing equal quantities of these three primary colour lights will produce white light. There are a number of ways of quantifying the amounts of each primary colour light present, which can represented by the values, r of the red component, g of the green component and b of the blue component; thus:

colour ¼ r þ g þ b

Use of these three additive primaries is called the RGB colour model. A simple colour space can be constructed by using Cartesian axes to represent the amount of the three primary colours, red, green and blue, while the diagonal represents the transformation from black to white (Figure 1.13a). Sections through this colour space allow one to represent colours by a planar ﬁgure. Such representations are called chromaticity diagrams. A simple example is given by taking the triangular sheet running diagonally through the cube normal to the black white diagonal and cutting the corners of the cube that represent pure red, green and blue. This produces a colour triangle (Figure 1.13b). Other colours can be

Figure 1.13 Colour spaces and chromaticity diagrams. (a) RGB colours represented by Cartesian axes, with black to white along the body diagonal. (b) A colour triangle, a section of (a) taken normal to the body diagonal passing through red, green and blue corners of the cube. A combination of the three primary colours at the vertices of the triangle will yield grey, but is shown white here. Other colours within the triangle (the gamut) can be represented by a point in the plane of the triangular system 31 Light and Colour specified by coordinates in the plane of the colour triangle. The location givenby the coordinates corresponds to the amounts r, g and b making up the colour. The coordinates which specify the case when the three primary colours are mixed in equal amounts will correspond to a shade of grey, but is usually represented by the colour white. The range of available colours that can be obtained by mixing lights corresponding to the threevertices is the gamut of colours available. Chromaticity diagrams generally represent hue and saturation, but not lightness (i.e. the grey tone), which must still be added as a third axis perpendicular to the chromaticity diagram if this information has to be displayed. The study of light mixing has been quantified by the Commission Internationale de l’Eclairage (CIE), which has, on a number of occasions, refined the rather simple colourtriangle concept so as toallow colourperceptions to be more accurately characterised. A colour is specified by a pair of x- and y-coordinates, which are derived from the r, g and b values noted above by the application of a standardized set of equations. In this representation, the triangular shape has been distorted into an outline something like a parabola, depending upon the way in which the x- and y-axes are plotted. A commonly encountered form of the CIE chromaticity diagram is that first proposed in 1931 (Figure 1.14a). The spectral colours are arranged around the outer edge of the shape and colours not seen in the spectrum, the purples and browns, are found to lie between the red and violet ends of the curve. The colours are fully saturated along the outer edge of the curve and become less and less saturated as the centre of the diagram is approached. Standard daylight white is represented by a point close to the coordinates x ¼ y ¼ 0.33, shown as W in Figures 1.14a, b. If a straight line is drawn through the point Wand extended to the boundaries of the curve, the pair of colours reached, when mixed, will give white light. For example (see Figure 1.14b), a line connects the colours red, of wavelength 700 nm and blue green, of wavelength 492 nm and passes through the point W. The proportions of the end colours red and blue green light needed to produce white light is given by the lever rule (Figure 1.14c):

amount of red light ¼ r=ðr þ cÞ amount of blue-green light ¼ c=ðr þ cÞ

Measurement shows that mixing red of wavelength 700 nm and blue green light of wavelength 492 nm in the proportions 39% red to 61% blue will produce white light. The colours at the ends of a line through the point W are called a complementary pair of colours. If one of these colours is subtracted from white light then the colour remaining is called the complementary colour to the ﬁrst. As with the colour triangle, all planar chromaticity diagrams represent hue and saturation, but not the exact value of lightness, which must still be added as a third axis perpendicular to the chromaticity diagram if this information has to be displayed. In general terms, therefore, the white region on the chromaticity diagram should be represented by grey, with white and black being extremes on the vertical axis perpendicular to the plane of the ﬁgure. Theaccuraterenditionofadditivecolorationisofprimeimportanceindisplays,suchastelevisionscreensand computer monitors. Additive coloration and the interconvertion between various colour models is most easily explored using a computer which has photography or drawing editing software installed. On most of these packages, seven or eight or so different colour models are available, including RGB and at least one CIE model. The coordinates of any colour are given and comparisons between several systems are rapidly made. The instructions and help facilities give full information upon these options and how they affect colour rendition. The confusion that colour blindness can cause is easily understood in terms of a chromaticity diagram. For example, Dalton had a lack of red receptors (Footnote 4). The CIE 1931 chromaticity diagram can be used to illustrate this. Any colour formed by mixing red with another colour, C, around the periphery of the curve will not be differentiated from any other colour along the line joining red to C. These lines show the loci of colour confusion (Figure 1.15). Other types of colour blindness will lead to other loci of colour confusion. Colour and the Optical Properties of Materials 32

(a) 520 green 0.8 510 540

560 0.6

y 500 yellow 580 orange 0.4 600 cyan 490 W red 700 0.2 480

400 violet 0.2 0.4 0.6 0.8 x

(b) 520 0.8 green 510 540

560 0.6

y 500 yellow 580

0.4 orange cyan 600 490 W red 700 0.2 480 violet 400 0.2 0.4 0.6 0.8 x

cyan white red 492 nm 700 nm Figure 1.14 The CIE 1931 chromaticity diagram. (a) The colours of the spectrum are arranged around a curved line and nonspectral colours fall on the line joining violet (400 nm) and red (700 nm). The ﬁgures marked around the outer edge of the curve denote the wavelength of the colour. Points within the area of the diagram represent colours formed by the additive mixing of light and can be speciﬁed by the appropriate x- and y-values. The point W represents white light. (b) A straight line through W links two complementary colours on the periphery of the diagram, in this example red and cyan. (c) The lever rule gives the proportions of complementary colours which are needed to create white light. In this example, the amount of red light is given by r/(r þ c) and the amount of cyan light by c/(r þ c) 33 Light and Colour

520 green 0.8 510 540

560 0.6 yellow y 500 580 orange 0.4 cyan 600 W 490 red 700 0.2 480

violet 400 0.2 0.4 0.6 0.8 x Figure 1.15 The dashed lines represent the loci of colour confusion for a person with red-defective vision plotted on the CIE 1931 chromaticity diagram. Because of a fault in the red perception, all colours on each line appear similar to the colour at the low wavelength extremity

1.13 The Interaction of Light with a Material

Colour is inherent in the light that leaves an emitting source; but most often before it reaches the eye it interacts with matter of many types: gases, liquids and solids. The colour observed is thus a function of both the source radiation and the interactions that have occurred. The way that light interacts with a material can be described in terms of scattering or absorption. To a ﬁrst approximation, scattering is well treated by assuming that the light behaves as an electromagnetic wave, while absorption is best treated in terms of photons. If the energy of the scattered wave/photon is the same as that of the incident wave/photon then the scattering is called elastic scattering, and otherwise inelastic scattering. For historical reasons, the term scattering itself, especially elastic scattering, is usually reserved for the interaction of light with randomly distributed small particles. Elastic scattering from a surface is normally called reﬂection, and elastic scattering into a transparent solid is called refraction. Scattering from ordered collections of small particles, or from small detail on largerobjects, is called diffraction. For the purposes of this book these terms are retained as they stand, although all are simply different aspects of scattering. All of these processes are wavelength dependent, and so can result in the production of coloured light from white light. Inelastic scattering arises when energy is transferred from the light photons to an absorption centre. Absorption is generally the term reserved for usewhen some or almost all of the incident radiation is taken up by the material and inelastic scattering when the changes are rather small. During absorption the energy is used to excite the component atoms or molecules that constitute the absorption centres into higher energy levels. Often, Colour and the Optical Properties of Materials 34

scattered light incident light

reflected light transmitted light

fluorescence

absorption centre

fluorescence centre scattering centre Figure 1.16 The interaction of light with a transparent material. The light can be reflected, absorbed or scattered. Some absorption centres are able to re-emit light as fluorescence or luminescence. All of the processes labelled are wavelength dependent and can lead to colour production the absorbed energy is manifested as a rise in temperature of the body. On occasion, some of this energy might be re-emitted as light, giving rise to fluorescence and related features. A material that does not absorb significantly is said to be transparent. Absorption may be minimal and transparency maximal for high-quality optical components over the visible spectrum, but no material is transparent over all wavelength ranges. Silicon, for example, appears ‘metallic’ over the visible spectrum but is transparent to infrared wavelengths. Absorption is wavelength dependent and an important source of colour production. It is often difficult experimentally to separate the relative roles that absorption and scattering play in the interaction of light with a material. As a beam of light passes through a material it gradually loses intensity, a process generally called attenuation (formerly extinction). Attenuation is due to the interaction of light with a material in two basic ways: scattering or absorption (Figure 1.16). When attenuation takes place in a homogeneous solid the amount of light transmitted by a semitransparent plate of thickness x is given by:

Ix ¼ Io expðaexÞð1:7Þ

5 1 where Ix is the irradiance leaving the plate, Io is the incident irradiance and ae (m ) is the (Napierian) linear attenuation coefficient (formerly extinction coefficient). Equation 1.7 is known as Lambert’s law or Beer’s law, although it was first clearly set out by Bouguer and should, by rights, be called Bouguer’s law. The

5 The symbol I is used for irradiance instead of E to avoid confusion with the use of E for energy throughout this book. See also Appendix 1.1. 35 Light and Colour

attenuation length is deﬁned as 1/ae. The amount of light removed from the beam is thus:

Irem ¼ IoIx ¼ IoIo expðaexÞ¼Io½1expðaexÞ

If the attenuation of the beam is solely due to absorption, then the attenuation coefficient is replaced by the (Napierian) linear absorption coefficient aa. Similarly, if the attenuation is solely due to scattering, then the attenuation coefficient is replaced by the (Napierian) linear scattering coefficient as. For nonhomogeneous solids these coefficients may vary with direction. Note that the degree of attenuation will vary significantly across the spectrum and the attenuation coefficient is not a constant. It is sometimes convenient, as when discussing the absorption of X-rays, to define a mass absorption coefficient m, which describes the decrease intransmitted irradiance through a homogeneous material of density r and thickness x:

Ix ¼ Io expðmrxÞ

In this case:

a m ¼ e r where m has units m2 kg 1 (in older literature cm2 g 1). Attenuation is often associated with the presence of chemical or physical ‘centres’, which may be atoms, molecules or larger particles, distributed throughout the bulk of a material. In the case of the mass absorption coefficient described above these are the totality of the atoms that make up the material itself. In this case, if the atoms in the material are supposed to absorb radiation independently of each other, then the mass absorption coefficient of the phase is simply related to the weight fraction of each atom species present. Thus, the mass absorption coefficient of a material M with a formula AxByCz is:

m ¼ð Þm þð Þm þð Þm M wt fraction A A wt fraction B B wt fraction C C

The weight fraction of each species is given by:

mass of A present xðm Þ wt fraction A ¼ ¼ A total mass xðmAÞþyðmBÞþzðmCÞ

and so on, where mA is the molar mass of species A, mB is the molar mass of species B and mC is the molar mass of species C. More often, extinction is associated with a dilute concentration of centres distributed throughout the bulk phase. In this case, the degree of extinction is often taken to be a function of the concentration of these centres. This is taken into account in the Beer Lambert or Beer Lambert Bouguer law: I log x ¼ecx Io Colour and the Optical Properties of Materials 36

where Ix is the irradiance after passage through a length of sample x, Io is the incident irradiance and c is the molar concentration (mol L 1, i.e. mol dm 3) of the active centres or species. The quantity e is called the molar (decadic) attenuation coefficient and has units6 of m2 mol 1. The attenuation coefficient has units of area and can, therefore, be regarded as an attenuation cross-section. In practical terms the units employed are often L mol 1 m 1 (i.e. dm3 mol 1 m 1). Writing 1 L as 0.001 m3, the molar attenuation coefficient can be expressed as 0.001 m2 mol 1 or 1 m2 mmol 1. The dimensionless product A ¼ ecx is called the absorbance (sometimes the optical density) and the ratio Ix/Io is the transmittance or transmissivity T. Thus, we can write:

logT ¼A

The Beer Lambert law ﬁnds use in the measurement of concentrations. For example, the clarity or otherwise of polluted air is often measured by comparing the irradiance of light at a certain time with the irradiance on a ﬁne day. These interactions with a material can be expressed thus:

Io ¼ Ir þ Is þ Ia þ It where Io is the incident irradiance, Ir is the amount reﬂected, Is is the amount scattered, Ia is the amount absorbed and It is the amount transmitted, or as:

1 ¼ R þ S þ A þ T where R is the fraction of light reﬂected, S is the fraction of light scattered, A is the fraction of light absorbed and T is the fraction of light transmitted and the quantities measured are the appropriate irradiance values. In good-quality optical materials the amount of light scattered and absorbed is small and it is often adequate to write:

Io ¼ Ir þ It or

1 ¼ R þ T

In a pure liquid the Beer Lambert law is often written in the form: I log x ¼ax Io where a (m 1) ¼ ec is the molar (decadic) attenuation (or absorption) coefﬁcient. The absorption will be due to molecular or atomic processes taking place in the pure medium.

6 Chemists frequently use the term molarity for concentration in mol L 1, given the symbol M. Thus, e is given the units M 1 m 1, or more often M 1 cm 1. To convert values of e in M 1 cm 1 to M 1 m 1, multiply the value by 100. 37 Light and Colour

Figure 1.17 Mediaeval stained glass window in Gloucester Cathedral, viewed from inside the building. [Reproduced with permission from Gloucester Cathedral www.gloucestercathedral.org.uk]

1.14 Subtractive Coloration

Absorption has been used for many centuries to produce colour. For example, the colour of stained glass and the colours seen in ordinary colour filters are examples of colour production in this way (Figure 1.17). The colours perceivedby the eyein which absorption and selectivereflection or transmission are important are said to be due to subtractive colour mixing. For example, the photosensitive pigments in green leaves preferentially absorb red and blue light and reflect more of the green component of the incident white light. Similarly, colour filters absorb some wavelengths strongly and transmit the remainder. Figure 1.18a shows the fraction of light transmitted as a function of wavelength for a commercial glass colour filter. The range of thevisible spectrum is indicated above the transmittance curve. The filter absorbs red light strongly and transmits violet and blue green light(Figure 1.18b). If the filter is held up tothe light it will look blue green. When it is viewed in reflected light it appears dark, as red light is absorbed and blue green passes through the film. This is the reason why stained glass windows in medieval churches look impressive when viewed inside the building, with light transmitted through the glass, yet often look dull when viewed from outside the building in reflected light. By analogy with additive coloration, one would expect to be able to combine three subtractive primary colours to produce the whole range of subtractive colours. These subtractive primary colours are: cyan, which absorbs red and transmits blue and green; magenta, which absorbs green and transmits blue and red; and yellow, which absorbs blue and transmits green and red. If the three subtractive primaries are mixed in equal amounts we obtain black, as one primary will absorb red, one will absorb green and one will absorb blue, thus removing the whole of the visible spectrum. Colour construction using these three subtractive primary colours is described as employing the CMY model, where the letters simply represent the initial letters of the colorants. If the wavelength range of light absorbed is rather small, then the colour remaining is called the complementary colour to that absorbed (Table 1.4). It is seen that the additive and subtractive primary colours are complementary colours. Colour printers use cyan, yellow and magenta dyes to produce the coloured images. These dyes are deposited upon white paper and absorb the appropriate subtractive primary colour. White light reflected from the dyes is depleted in these colours and yields the appropriate toned image by subtractive coloration. Although the Colour and the Optical Properties of Materials 38

1.0 blue red

visible

0.5 Fractional transmittance

(a) 300 500 700 Wavelength / nm

filter

blue-green white light light

(b)

Figure 1.18 (a) The fractional transmittance of a commercial blue colour filter. About three-quarters of the blue light incident on the filter is transmitted, but most red light is absorbed. (b) When the filter is viewed in transmitted white light it will appear blue–green

Table 1.4 Complementary colours

Wavelength/nm Colour absorbed Complementary colour 400 435 Violet Yellow green a b 435 480 Blue Yellow 480 490 Blue green Orange 490 500 Green blue Red a b 500 560 Green Magenta 560 580 Yellow green Violet 580 595 Yellow Blue 595 605 Orange Blue green a b 605 700 Red Cyan

a Additive primary colours. b Subtractive primary colours. 39 Light and Colour overlap of cyan, yellow and magenta produces black, this tone is often not dark enough for many representations. Printers, therefore, often add black to the trio. This system of colour production is known as the CYMK model of colour formation, where the letter K stands for the black component. Although these four colours are satisfactory for many colour printing applications, more hues, intermediate between the CYMK set, are used to obtain more accurate colour rendition, in, for example, high-quality art reproductions.

1.15 Electronic ‘Paper’

Paper is an extremely convenient way of displaying information using subtractive coloration, but once a page is printed it is permanent. Electronic paper, with the advantages of a printed page, but the flexibility of electronic erase and rewrite has been pursued for over 30 years. As of 2000, e-book readers, which are rigid units displaying one paper-like page at a time, have been increasingly available. There are two aspects to electronic paper. In the first, electronic ‘ink’ must be developed that will retain the display indefinitely but is erasable at will. At least for black-and-white displays this has been accomplished. The second is the production of a flexible page that can support the electronic circuitry needed to drive the display. In this section the characteristics of the ‘ink’ are the main focus of attention, as this is the aspect that impinges upon the topic of colour. The first electronic paper, using the Gyricon process, consisted of small polyethylene spheres of approximately 90 mm diameter, coloured white on one hemisphere and black on the other. Thewhite part held a positive charge and the black portion a negative charge, due to additives to the polymers used. These spheres were embedded in a transparent silicone film and the sheets were immersed in clear oil. This penetrated the sheets and coated the beads, so that they were effectively encapsulated in a bubble of oil. The application of a negative charge to an electrode on the surface will attract the positively charged white side facing one side of the ‘page’. In this way pixels of the display could be made black or white at will (Figure 1.19a). Rearranging the applied voltage allows the image to be erased and rewritten. The e-ink process is rather similar but uses the movement of charged particles in an electric field, the process of electrophoresis. Once again, small polymer capsules containing submicrometre particles of white titanium dioxide, TiO2, holding a negative charge due to appropriate surfactants, and black particles holding a positive charge are central to the system. The microspheres also contain a nonviscous liquid and are embedded in a clear plastic film. A charge applied to surface electrodes will attract white or black particles depending upon the polarity of the electrodes. Reversal of the charge on the electrodes reverses the particles that are attracted and the area will swap colour (Figure 1.19b). Erasure and rewriting is carried out as before. Naturally, the use of polymer spheres to contain the black and white particles is not mandatory, and any cell structure could be used. The device also becomes simpler if the black particles are replaced by a dark-coloured fluid. The white particles are then the only active species present. When attracted to a surface the appropriate pixel looks white and when not attracted the dark fluid is seen. The colour of the pixels is due to absorption and scattering. Titanium dioxide is a well-known white scatterer (Section 5.7) and the dark colour is simply absorption of the incident light by the dye present. The system can be made into a colour display by putting red-, green- and blue-coloured filters in front of the electrodes (Figure 1.19c). A white pixel will now become a coloured subpixel corresponding to one of the colours. The electrodes used to control the display can be a simple passive array of vertical strips on one face of the device and horizontal strips on the reverse face. Application of charges to appropriate columns and rows ensures that pixels can be made black or white as required. The active matrix method of control, as used in flat- panel television, in which a transistor controls each pixel, is also widely used. An advantage of these displays is that once the page has been created, no further electrical input is needed until the page is rewritten. Of course, Colour and the Optical Properties of Materials 40

++_ ___ + + _ __++ _ ++ (a)

_ + +

(b) + _

–––+ + + colour filter layer +

– + –– + –_ + (c) Figure 1.19 Electronic paper displays: (a) the rotating sphere Gyracon system; (b) the electrophoretic e-ink system; (c) coloured ﬁlters allow for a full colour display to be achieved the requirements are less demanding for a rigid e-book than for a portable and ﬂexible sheet-like page, which has still to achieve widespread commercialization.

1.16 Appearance and Transparency

Scattering and absorption give rise to the world of colour around us (Figure 1.20). Even small changes in the relative amounts of each wavelength band present in a light beam will contribute significantly to colour and appearance. A striking example of this is the blue sky. Blue sky is so coloured because of light scattering (see Chapter 5). However, blue sky contains all of the wavelengths of the spectrum something easily proved by passing the light through a prism. The sky appears blue because the balance in the various colours has been tipped slightly in favour of the blue end of the visible spectrum. The appearance of an object will depend on a number of factors, especially on roughness and surface texture. These will alter the reflectivity of the surface considerably. If the surface is smooth then the reflection is said to be specular, while if the surface is rough then the reflection is diffuse (Figure 1.21a). The diffuse reflection component increases with surface roughness at the expense of the specular component, so that a finely ground powder shows only diffuse reflection. The gloss of a surface is a measure of the relative amounts of diffuse to specular reflections. Glossy surfaces have a large specular component. As well as diffuse reflection, subsurface scattering is of considerable importance in modifying the appearance of a surface. This is particularly so when the surface is composed of layers with different optical properties, such as skin. Controlling these forms of scattering and reflection are of great importance to the cosmetics industry, and imitating them is vital to both artists and personnel involved in the representation of skin tones in computer-generated images. 41 Light and Colour

Figure 1.20 A moorland scene displaying colours due to scattering (blue sky), reﬂection (the blue stream) and absorption (the green–browns of grass and soil)

Closely related to this is the property of transparency or invisibility in animals. Invisibility confers obvious advantages to both predator and prey in the living world. It is not surprising, therefore, that many marine animals almost achieve this object. To attain invisibility, the interactions of light with a material described above must be bypassed. That is to say, reflection and refraction at surfaces and scattering and absorption from internal centres need to be suppressed. Reflection and refraction at the surface of the animal’s body can be substantially reduced by making the refractive indices on both sides of the boundary the same (Chapters 2 and 3). For many marine animals, including numerous species of zooplankton, jelly fish and similar creatures, the inner body fluid is essentially watery, and reflection and refraction are virtually eliminated. This alone serves to make the animals virtually invisible.7 However, any inhomogeneities in the tissues and membranes will act so as to scatter light and render the animal visible to a greater or lesser degree. Air pockets are particularly problematic. For instance, small bubbles of air in water are easily visible in ordinary light and shine like silver spheres. Pigments, which colour by absorption, cannot be totally avoided. The photoreceptors of the eye are pigments which absorb visible radiation (Section 1.10). Similarly, the prey of the animal, once consumed, will be visible in the gut, unless this matches the surroundings in both transparency and refractive index. Thus, although many marine animals can be extremely difficult to spot, and may be termed invisible for all practical purposes, some traces will remain visible. Solids and liquids cannot be manipulated so that their refractive index matches that of air, but can be made with a refractive index that matches that of a liquid. A transparent solid immersed in a liquid of the same refractive index will be invisible. For many solids, internal surfaces are a major cause of loss of transparency.

7 This is the basis of the famous story The Invisible Man by H. G. Wells, published in 1897. Colour and the Optical Properties of Materials 42

diffuse reflection specular reflection incident beam

rough surface

(a)

diffuse reflection specular incident beam reflection

opal glass

specular diffuse transmission transmission (b) Figure 1.21 (a) Reflection of light from a rough surface consists of two components, diffuse reflection and specular reflection. The ratio of diffuse reflection to specular reflection increases as the surface roughness increases. The ratio is an indication of surface gloss. (b) The passage of light through a translucent material containing many scattering centres gives rise to both surface reflection and transmitted light with diffuse and specular components

Glass, the best known of transparent solids, is, in effect, in the liquid state, and no internal boundaries occur. However, it is relatively simple to cause a glass to crystallize and the ensuing tiny crystallites act as scattering centres. The resulting scattering, which can contain diffuse and specular components, renders the material nontransparent although the solid transmits a certain amount of the incident light. Such materials are termed translucent. The light emerging from a translucent material will also contain a diffuse and specular component (Figure 1.21b). Translucency is a desirable property of ﬁne porcelain, which consists of crystallites of mullite (Al6Si2O13) dispersed throughout a glassy matrix. More opaque glasses, such as opal glasses, are deliberately made with large numbers of scattering centres present. The resultant scattering renders the material white because the scattering affects all wavelengths of the incident light equally. Similarly, most plastics as fabricated are noncrystalline and have no internal boundaries, rendering them transparent. If these contain impurities, inhomogeneities or polymer crystallites they become translucent and take on a slightly milky appearance. Non-glassy solids are mainly composed of polycrystalline aggregates or ‘grains’. The grain boundaries between each crystallite scatter light, and any impurity phases that exist in the matrix or the grain boundary regions enhance this effect, so that polycrystalline solids are invariably opaque. However, it is of considerable beneﬁt to make such materials transparent. This can be achieved by careful processing that achieves a high 43 Light and Colour density, so that internal pores and bubbles of gas are eliminated, and produces a solid composed of small, evenly sized crystallites with no impurity grain-boundary phases present. In this way, transparent refractory ceramics such as alumina (Al2O3), aluminium oxynitride (Al23O27N5) and SiAlONs (materials occurring in the SiO2 Al2O3 Si4N3 AlN system) have been produced. These and similar materials have uses as lamp housings and windows which need to be stable in air to temperatures of 2000 C or more. In addition, these are hard and durable ceramics and are favoured for applications such as specialist optical windows and domes where resistance to abrasion and erosion are important selection criteria.

Appendix A1.1 Deﬁnitions, Units and Conversion Factors

A1.1.1 Constants, conversion factors and energy Constants The important constants for light are: velocity of light in vacuum c 2.99792 108 ms 1 Planck constant h 6.62608 10 34 Js 23 1 Boltzmann constant kB 1.38066 10 JK

Conversion Factors E (J) ¼ E (eV) 1.60219 10 19 E (J) ¼ E (cm 1) 1.98645 10 23 E (eV) ¼ E (cm 1) 1.23987 10 4 l (A) ¼ l (nm) 10 l (nm) ¼ l (mm) 1000 l (nm) ¼ 1239.9/E (eV) l (nm) ¼ 198 645 10 21/E (J) l ðnmÞ¼107=n ðcm 1Þ

Energy The SI unit of energy is the joule (J). Awidevariety of energy units are used in the literature connected with light apart from the joule. A common nonstandard unit of energy in atomic work is the electron volt (eV). Spectroscopy often uses energy values given in cm 1. These are not energy values at all really, but E/hc values. To convert ‘energy’ in cm 1 to joules, multiply the value in cm 1 by h (J s) and c (cm s 1); see Conversion Factors above.

A1.1.2 Waves Waves The wave equation, Equation 1.2, is a one-dimensional continuous harmonic wave that represents the electric ﬁeld vector E:

E¼E0 cos½ð2p=lÞðxvtÞ ðA1:1Þ Colour and the Optical Properties of Materials 44

E is the magnitude of the electric ﬁeld vector at position x and time t, E0 is the amplitude of the wave, l is the wavelength of the wave, ½ð2p=lÞðxvtÞ is the phase of the wave (radians), v is the speed at which any point on the wave, say a peak or a trough, travels in the positive x direction, and is called the phase speed or phase velocity. The velocity of an electromagnetic wave in vacuum (the speed of light) has the symbol c. The relationships described below allow the equation to be written in other equivalent forms. Those most frequently met are: 1. a standing (non-travelling) wave:

E¼E0 cos½ð2p=lÞx

2. a wave travelling in the negative x direction:

E¼E0 cos½ð2p=lÞðx þ vtÞ

3. a wave travelling in the positive x direction, where o is the angular frequency:

E¼E0 cos½ð2px=lÞot

4. a wave travelling in the negative x direction, where o is the angular frequency:

E¼E0 cos½ð2px=lÞþot

Frequency The temporal frequency n of light (the number of waves that pass a point per second) has units of cycles per second, hertz (Hz) or s 1. It is usually just called the frequency. The reciprocal of the temporal frequency, 1/n,is the temporal period t, which is the amount of time for a complete wave oscillation to pass a stationary observer at a ﬁxed value of x. The angular (temporal) frequency o of a wave is given by: p o ¼ 2 ¼ pn 1 t 2 units: rad s

Using the relationship c ¼ ln gives o ¼ 2pc/l.

Wavelength The wavelength l. is the spatial period of the wave the distance over which the wave subsequently repeats itself. In wavelength designations concerning light, nanometre (nm) is the preferred unit, but a commonly used nonstandard unit, especially in X-ray diffraction, is the angstrom€ (A), 10 10 m. To convert between units, see Conversion factors above.

Wavelength and Energy Planck’s law (E ¼ hn ¼ hc/l ¼ ho/2p ¼ ho) relates energy to wavelength. To convert between units, see Conversion factors above. 45 Light and Colour

Wavenumber The wavenumber is the reciprocal of the spatial period of the wave (the number of waves per unit length) and so is the reciprocal of the wavelength, 1/l. The wavenumber is given the symbol s when the light traverses a transparent medium or n when in a vacuum. In spectroscopy it is often given units of cm 1.

sðnÞ¼1 l

Using Planck’s law, in a vacuum:

¼ n ¼ hc ¼ n E h l hc E ¼ n hc

Similar equations can be written for light in a substance, by replacing c by the velocity v in the medium and replacing n with s. Spectroscopy often uses wavenumbers (units: cm 1). To convert these to wavelength (units: nm):

107 l ðnmÞ¼ n ðcm 1Þ

In physics, the magnitude of the propagation vector or wave vector k is called the propagation number or wavenumber, given by k ¼ 2p/l. (By analogy with temporal frequency and temporal angular frequency, it might be better to call k ¼ 2p/l the angular spatial frequency to avoid confusion.) Additionally, physics also uses k ¼ 1/l for the wavenumber, omitting the factor 2p. To avoid confusion, the wave vector will be written as 2p/l or 1/l rather than k.

A1.1.3 SI units associated with radiation and light

There are two parallel sets of units in use for the measurement of radiation and light. Photometric units measure the perception of a light as it appears to the eyeof an average observer. Radiometric units measure the amount of electromagnetic radiation, including light, in terms of absolute quantities, without any reference to the eye. The difference can be understood by considering the light output of four small light-emitting diodes (LEDs), one infrared, one deep red, with an emission at 670 nm, one green with an emission at 555 nm and one blue with an emission at 490 nm. These may all emit exactly the same absolute power (measured in radiometric units, say 5 mW), but the green light will appear ‘brighter’ than the other two visible LEDs because the eye is more sensitive to green than red or blue. Calculation shows that the visible outputs will be: green, approximately 3.4 lm; blue, approximately 0.75 lm; red, approximately 0.1 lm; infrared, 0 lm. The green LED will appear about 31 times brighter than the red LED, and the blue LED about seven times brighter than the red LED. The infrared-emitting LED will be invisible to the eye and will not register at all in terms of photometric units, although it still emits the same amount of power as the visible ones. Clearly, photometric units are of importance in the design of displays and lighting, whereas radiometric units are of more importance when comparing the energy requirements of the same structures. Although the two sets of units are analogous, as set out in Table A1.1, because they measure different aspects of light, they cannot be trivially interchanged in this regime. Table A1.1 Units used in radiometry and photometry

Radiometry Photometry

Name, symbol Comments Units Name, symbol Comments Units Radiant power, Rate of flow of energy W Luminous power, Rate of flow of luminous energy lumen (lm) radiant flux, F, P emitted by a source luminous flux, emitted by a source (cd sr) F, Fv Radiant intensity The power of an emitting Wsr 1 Luminous Light emitted from a source per unit candela 1 I ¼ dF/d source per unit solid angle intensity, Iv solid angle; SI base unit (cd) ¼ lm sr 2 1 2 Radiance, L ¼ Radiant power per unit area Wm sr Luminance, Lv A measure of ‘brightness’; luminous cd m (nit!) d2F/(dA d) per unit solid angle. intensity of a light emitting source Radiant intensity of a per unit area of source; may vary radiating source per unit over the source surface surface area. 2 2 Irradiance E (I) ¼ Radiant power incident Wm Illuminance, Ev A measure of illumination; the lux (lm m ) dF/dA upon a unit area of a luminous flux falling on a surface surface. per unit area 2 2

(Radiant) Exitance Radiant power emitted by a Wm Luminous Luminous ﬂux emitted from a surface lux (lm m ) Materials of Properties Optical the and Colour M ¼ dF/dA surface per unit area exitance, Mv

Flux is the amount of something flowing through a specified surface per unit time. Luminous flux or luminous power F, unit lumen (lm): 1 lm is the amount of luminous flux passing in 1 s through a unit solid angle emitted by a point source of 1 cd. The total luminous flux of such a point source is 4p lumens. 12 Luminous intensity Iv, unit candela (cd): 1 cd is the photometric measurement of luminous intensity in a given direction of a source that emits monochromatic radiation of frequency 540 10 Hz and that has a radiant intensity in that direction of (1/683) watts per steradian. One square metre of a black body at 2042 K emits 600 000 cd. Radiance L, units W m 2 sr 1: the radiance is the incoming radiation collected from a small angle of surroundings (measured in steradians) as if, for example, the detector is at the bottom of a tube. The units of radiance are energy per unit area per unit solid angle, W m 2 sr 1. The radiance is direction sensitive – the value recorded depends upon the direction in which the tube is pointing. Irradiance E or I, unit W m 2: the radiometric term irradiance is the total energy that a detector ‘sees’ from a hemisphere of surroundings. The preferred symbol for irradiance is E, but because of the use of E for energy, and of E for the amplitude of an electromagnetic wave, it is less confusing here to use the symbol I. 2 Illuminance Ev, unit lux (lx): this is the photometric analogue of irradiance, being the total luminous flux incident upon unit area of a surface, with units of lux ¼ lm m . The photometric term illuminance has replaced the term brightness. Radiant exitance or radiant emittance M, unit W m 2: the amount of electromagnetic radiation leaving a surface is described by the radiometric term (radiant) exitance. The exitance is the opposite of the irradiance, as it measures the total energy emitted by a surface into a hemisphere of the surroundings. The exitance has the same units as irradiance. Luminous exitance or luminous emittance Mv, unit lux (lx): this is the photometric analogue of the radiometric radiant exitance. Spectral units. These give the distribution of the quantity under discussion with respect to the wavelength or frequency of the radiation. For example, the spectral irradiance takes the form irradiance per unit wavelength, written El, or irradiance per unit frequency En. The units of spectral quantities must contain the units of wavelength or frequency as appropriate. Thus, the units of spectral irradiance are W m 2 m 1 ¼ Wm 3. 46 47 Light and Colour

Further Reading

The following ﬁvebookscontain avast amount ofmaterial ofrelevance tothewhole ofthisbook.The two books by Bohren present material in a nonmathematical format and will repay repeated reading. E. Hecht, Optics, 4th edition, Addison-Wesley, San Francisco, 2002. K. Nassau, The Physics and Chemistry of Color, 2nd edition, Wiley-Interscience, New York, 2001, Chapters 1 and 2 and Appendix A. C. F. Bohren, Clouds in a Glass of Beer, Dover, New York, 2001 (originally published by John Wiley and Sons, Inc., New York, 1987). C. F. Bohren, What Light Through Yonder Window Breaks? Dover, New York, 2006 (originally published by John Wiley and Sons, Inc., New York, 1991). B. E. E. Saleh, M. C. Teich, Fundamentals of Photonics, John Wiley and Sons, Inc., New York, 1991. Colour, from the point of view of artist’s pigments, is the subject of V. Findlay, Colour; Travels through the Paintbox, Folio, London, 2009. Goethe’s Theory of Colour, written 1808, published 1810, gives an interesting historical view of colour. It is included in D. Miller, (ed. and trans.), Goethe: Scientiﬁc Studies, Suhrkamp, New York, 1988 (Goethe Edition Vol. XII). A clear discussion of radiometric and photometric units is given by J. M. Palmer (2003) to be found at http:// www.optics.arizona.edu/Palmer/rpfaq/rpfaq.htm. Also see C. F. Bohren, E. C. Clothiaux, Fundamentals of Atmospheric Radiation, Wiley VCH, Weinheim, 2006, Chapter 4. The electromagnetic theory of radiation is clearly set out in M. Kotlarchyk, Electromagnetic radiation and interactions with matter, in Encyclopedia of Imaging Science and Technology, J. P. Hornak (ed.), Wiley-Interscience, 2002. N. Braithwaite (ed.), Electromagnetism, Book 3, Electromagnetic Waves, The Open University, Milton Keynes, 2006. Light described in terms of quantum electrodynamics is explained lucidly and nonmathematically in R. P. Feynman, QED: The Strange Theory of Light and Matter, Princeton University Press, Princeton, 1985. The fascinating history of the theories of light is given by G. N. Cantor, Optics after Newton, Manchester University Press, Manchester, 1983. A comparison between the wave and particle explanation of the photoelectron effect and profound discussions of the relationship between particle and wave theories of atomic physics are given by D. Bohm, Quantum Theory, Prentice-Hall, Englewood Cliffs, NJ, 1951. A detailed discussion of the solar spectrum and related topics is given by C. F. Bohren, E. E. Clothiaux, Fundamentals of Atmospheric Radiation, Wiley VCH, Weinheim, 2006, Chapter 1. D. K. Lynch, W. Livingston, Color in Light and Nature, Cambridge University Press, Cambridge, 1995, Chapters 2 and 7. The (coincidental) relationship between the sensitivity of the human eye and the solar spectrum is discussed by B. H. Stoffer, D. K. Lynch, Am. J. Phys. 67, 946 958 (1999). For discussions of the molecular basis of vision, see D. M. Hunt, L. S. Carvalho, J. A. Cowing, W. L. Davies, Phil. Trans. R. Soc. Lond. Ser.B364, 2941 2945 (2009). Colour and the Optical Properties of Materials 48

For a discussion of bacteriorhodopsin, see J. Whitford, Proteins, Structure and Function, John Wiley and Sons, Ltd, Chichester, 2005, pp. 114 119. The evolution of primate colour vision is detailed by G. H. Jacobs, J. Nathans, Sci. Am. 300 (April), 40 47 (2009). The complexity of the retina and much information about vision in specialist circumstances is to be found by consulting S. Temple, N. S. Hart, N. J. Marshall, S. Collin, Proc. R. Soc. Lond. Ser.B277, 2607 2615 (2010). Many aspects of vision and the interpretation of visual images, including optical illusions, are detailed in the series of articles by J. C. Russ, Seeing the scientiﬁc image, Proc. R. Microscop. Soc. 39 (2004); Part 1: 97 114; Part 2: 179 193; Part 3: 267 281. The complexities of analysing colour and descriptions of the construction and use of chromaticity diagrams are detailed in the following sources. A largenumber of articles concerning colour, colour theory, colour systems and colour spaces will be found on Wikipedia (http://en.wikipedia.org/wiki/). Up-to-date details of colour and colour reproduction will be found in the Instructions and Help functions for computer drawing and photograph editing software, many of which are available: typically as in manuals for Nikon Coolscan, Coreldraw, Photoshop and so on. Other sources are http://www.efg2.com (this site has programs for the display and representation of chromaticity diagrams, colour mixing and many other topics of relevance to the material in this chapter). R. McDonald, Colour Physics for Industry, 2nd edition, Society of Dyers and Colourists, Bradford, UK, 1997. F. Grum, C. J. Bartleson, Colour Measurement, Academic Press, New York, 1980. R. Jackson, L. MacDonald, K. Freeman, Computer Generated Colour, John Wiley and Sons, Ltd, Chichester, 1994. Other interesting sources on colour and colour perception are as follows: Animal colour patterns J. A. Endler, J. Linn. Soc. 41, 315 352 (1990).

Reviews of transparency in biological tissues S. Johnsen, E. A. Widder, J. Theor. Biol. 199, 181 198 (1999). S. Johnsen, Sci. Am. 282 (February), 62 71 (2000).

Texture and computer modelling of surfaces J. Dorsey, P. Hanrahan, Sci. Am. 282 (February), 46 53 (2000) and references cited therein. There are a number of demonstrations of relevance to this chapter, including diffuse versus specular reﬂection, available at http://demonstrations.wolfram.com/index.html. 2 Colours Due to Refraction and Dispersion

. Why are images of objects in water displaced? . How do rainbows form? . What is a negative index material (NIM)?

Many of the most beautiful colours arise simply by the passage of a beam of ordinary white light through a transparent material. Rainbows provide a familiar example. These effects are due to refraction and dispersion by the material. In this chapter, these rather straightforward features are described. The effect of the polarisation of the light is ignored for the present, although this feature is of importance, particularly for refraction within crystalline solids or when the beam is especially powerful. These complexities are considered later.

2.1 Refraction and the Refractive Index of a Material

When light travelling through the air enters a transparent medium, say a sheet of glass or a pool of clear water, it appears to bend. This is called refraction. The effect of refraction is familiar to most. A stick will appear bent towards the surface when dipped into water (Figure 2.1a and b). Similarly, the bottom of a swimming pool always seems closer to the surface than it really is. Kingfishers and other birds that catch fish by diving into rivers must allow for this effect and aim below the object that they apparently see in order to hit the target (Figure 2.1c). An equally complex problem is faced by Archer fish. These animals capture insect prey by spitting a jet of water onto overhanging vegetation when an insect is present, knocking the prey into the water below. The eyes of the Archer fish remain below the water during a capture attempt, and refraction as well as associated colour changes (Section 2.6) must all be included in the ‘equation’ for the water jet trajectory. The magnitude of the deviation as a light ray passes across the boundary into a transparent substance is quantified by the refractive index (or index of refraction) of the material. For all practical purposes involving

Colour and the Optical Properties of Materials Richard J. D. Tilley 2011 John Wiley & Sons, Ltd Colour and the Optical Properties of Materials 50

(b) observer

air A

water C

air

water

Figure 2.1 Refraction at an air–water surface. (a) A half-immersed pencil seems to bend upwards; (b) the cause of the apparent displacement is that rays of light reflected from B appear to come from C; (c) an object such as a fish in a pool appears to be nearer the surface than it is because light rays reflected from the fish at B appear to originate from C 51 Colours Due to Refraction and Dispersion light, the refractive index of a transparent solid is positive.1 To treat the interaction of light with a wide range of materials, not just transparent ones but also those that are semitransparent or opaque, it is necessary to consider the refractive index as a complex number.2 In this case, the complex refractive index N is written in the form:

N ¼ n þ ik where n is the refractive index and k is the absorption index or extinction coefﬁcient.3 (Note that the complex refractive index is also written N ¼ n ik. The formalism chosen depends upon the way in which the light wave is described mathematically.) The pair of terms n and k are called the optical constants of a material. (Both of these terms vary considerably with wavelength and, in practice, are by no means constant!) The real part of the complex refractive index, n, accounts for the interaction of light with the nonabsorbing part of the medium and the imaginary part of the complex refractive index, k, represents the absorptive properties of the medium through which the light travels. For a completely nonabsorbing material (at least over the wavelength range of interest), the absorption index k is zero. The behaviour of these materials is expressed simply in terms of n, the ‘ordinary’ refractive index. In cases where absorption is present, the absorption coefﬁcient aa is related to k by:

p a ¼ 4 k ð : Þ a l 1 7

Materials that absorb over part of the electromagnetic spectrum are frequently transparent in other parts. Silicon, for example, appears to be metallic in visible light but is transparent in the infrared. Water strongly absorbs over much of the electromagnetic spectrum, making it a potent ‘greenhouse gas’, although it is transparent in the visible. The refractive index of the oxide ceria, CeO2, is close to 2.35 over the visible spectrum. It is a transparent phase with k ¼ 0. At the onset of the ultraviolet, this changes rapidly and the value of k reaches approximately 1.4 at 310 nm. This property makes ceria a possible component of sunscreen creams. The magnitude of the ray deviation when light enters a transparent medium is related to the index of refraction by:

sin n ¼ 1 ð2:1Þ sin2

1 being called the angle of incidence and 2 the angle of refraction (Figure 2.2). This equation is known as Snel’s law.4 The plane of incidence is the plane containing the incident ray and the normal to the surface. The

1 There is no theoretical reason why the refractive index should always be positive, and research into negative refractive index materials is important. This is discussed in Section 2.10. 2 Complex numbers are simply an ordered pair of numbers. One part is called the ‘real’ part and the other the ‘imaginary’ part. This terminology frequently gives rise to confusion or a lack of understanding. In fact, one part could easily be called the ‘red’ part and the other the ‘blue’ part instead of the ‘real’ part and the ‘imaginary’ part. In the present case, the real part gives the interaction with the transparent aspect of the medium and the imaginary part describes the interaction with the absorptive part of the medium. The use of the complex number formalism allows both aspects of the phenomenon to be analysed mathematically simultaneously. 3 Note that the symbol k is used both in the complex refractive index, for the wavenumber of an electromagnetic wave (see Appendix A1.1), for the Boltzmann constant and for the refractive coefﬁcient (Section 2.4). In this book, the use of k is restricted to the absorption index. The wavenumber will be written 1/l or 2p/l as appropriate, the Boltzmann constant as kB and the refractive coefﬁcient as kr. 4 The originator of this ‘law’ was Willibrord Snel van Royen (1591 1626). The spelling ‘Snell’ is incorrect, but well established. Colour and the Optical Properties of Materials 52

θ 1

air

glass

θ 2

Figure 2.2 The refraction of a beam of light as it enters a block of a transparent medium, such as glass, with a high refractive index from a medium of low refractive index, such as air. The light beam appears to bend at the interface by an amount given by Snel’s law above equation is a special case of the more general relation:

sin n 1 ¼ 2 ð2:2Þ sin2 n1 for light passing from a medium of refractive index n1 to one of refractive index n2. In effect, the refractive index of a transparent material is a manifestation of the fact that the light wave is slowed down on entering a transparent material. This is due to the interaction of the light with the electrons around the atoms which make up the solid. The absolute refractive index is given by:

¼ c ð : Þ n v 2 3 where c is the velocity of light in a vacuum and v the velocity of light in the medium. The frequency of the light does not alter when it enters a transparent medium, and because of the relationship between the velocity v and frequency n, that is:

nl ¼ v it is possible to write:

c l n ¼ ¼ 0 ð2:4Þ v ls where l0 is the wavelength of the light wave in a vacuum (which is close to that for air) and ls is the wavelength in the transparent substance. It is thus seen that light has a smaller wavelength in a transparent material than in air or a vacuum (Figure 2.3a). 53 Colours Due to Refraction and Dispersion

(a)

n = 1.0 n = 2.0 n = 1.0

d d d

(b)

n2 out of step d (c)

n1 n2 n1 n3 n2

d1 d 2 d 3 d 4 d 5 Figure 2.3 The effect of the refractive index on the wavelength of light. (a) The wavelength in a medium of refractive index 2.0 is half that in a medium of refractive index 1.0. (b) A wave divided into two and passing through materials of refractive index n1 and n2 and of thickness d will be out of step on emergence. (c) The optical thickness of a series of slabs is the sum of the optical thickness, n1d1, etc. of the optical thickness of all of the slabs

This can introduce confusion when the path of light rays through different materials has to be compared. To overcome the difﬁculty it is useful to deﬁne the optical path or optical thickness [d] and distinguish it from the real or physical thickness of a material d. The relationship is given by:

½d¼nd ð2:5Þ

A slab of thickness d and refractive index 2n (optical thickness 2nd) will ‘contain’ twice as many wavelengths as a slab of thickness d and refractive index n (optical thickness nd) (Figure 2.3a). In effect, the optical thickness is a measure of dimensions in terms of the wavelength of the light passing through it. Two different materials with the same optical thickness contain the same number of wavelengths of the light which traverses each of Colour and the Optical Properties of Materials 54

them. Similarly, if a beam of light is divided so that one part enters a slab of thickness d and refractive index n1 while the other part enters a slab of thickness d and refractive index n2, on emerging from the slabs any point on wave 1, say a crest, will have travelled n1d while the equivalent crest on wave 2 will have travelled n2d.In general, the two waves will be out of step by (jn1 n2j)d, where only the absolute value of the difference in refractive indices is important (Figure 2.3b). For several transparent materials traversed in sequence (Figure 2.3c):

Xn ½d¼n1d1 þ n2d2 þ n3d3 þ ¼ nidi ð2:6Þ i¼1

In many crystalline materials the index of refraction varies with the direction of the beam of light. This is taken further in Chapter 4. In the present chapter only those materials where the index of refraction is independent of direction are considered. These are said to be optically isotropic and are typiﬁed by gases, liquids, glasses and crystals with cubic symmetry. The refractive index can also vary locally in intense light beams, as when a laser beam enters a transparent solid. This gives rise to a nonlinear refractive index, which is most conveniently discussed in the context of crystal optics (Chapter 4).

2.2 Total Internal Reﬂection

2.2.1 Refraction at an interface

When light passes from a higher refractiveindex material, such as glass, to one of lower refractiveindex, such as air, the refraction causes the emerging ray to bend towards the interface (Figure 2.4a). As the angle of incidence 1 at which the ray approaches the surface increases, the angle of the emerging ray 2 increases and the ray approaches the surface (Figure 2.4b). At the critical angle c the emerging ray actually travels exactly along the surface (Figure 2.4c). If c is exceeded, then no light escapes and it behaves as if it were reﬂected from the undersurface (Figure 2.4d). This effect is called total internal reﬂection. The light is trapped in the high refractive index medium. The critical angle is given by:

n2 n ðlowÞ sinc ¼ ¼ ð2:7Þ n1 n ðhighÞ

which follows from Equation 2.2 when 2 ¼ 90 . Total internal reflection is not an ‘all-or-nothing’ process. When the incident light falls onto the interface at normal incidence it is completely transmitted. As the angle of incidence approaches the critical angle, more and more light is reflected back into the medium of higher refractive index and less and less is transmitted. At the critical angle, this shift is complete and everything is reflected (but see the following section). Total internal reflection from the glass air interface is used in prismatic binoculars and single-lens-reflex cameras to ‘reflect’ light and to channel light in optical fibres (Section 2.9). It is the reason why a swimmer underwater will see the air surface as a bright ‘hole’ in a surrounding dark continuum.

2.2.2 Evanescent waves

When waves encounter a barrier they often get around it in one way or another. In the present context, a light wave that is totally internally reﬂected at an interface can ‘leak’ across it, although such an occurrence is 55 Colours Due to Refraction and Dispersion

(a)

low refractive θ index n2 2

high refractive θ 1 index n1

(b)

θ 2 n2

n1 θ 1

(c)

θ 2 n2

θ θ 1 = c

(d) n2 n 1 θ θ 1 1

Figure 2.4 (a, b) When light passes from a medium of high refractive index (such as glass) to one of low refractive index (such as air) the transmitted portion will be refracted into a path which lies closer to the interface between the materials while the remainder is reflected below the interface. The amount transmitted gradually decreases and the amount of reflected light gradually increases as the angle of incidence increases. (c) When the angle u of incidence reaches the critical angle c the vanishingly weak transmitted ray will emerge along the surface itself. (d) For angles of incidence greater than the critical angle, total internal reflection will occur and all the light is reflected Colour and the Optical Properties of Materials 56

(a) evanescent wave n2

n1 θ θ 1 1

(b)

θ 1 n1

n2 evanescent wave n 1 θ θ 1 1

coupling region

output signal 2

(d) output signal 1

output signal 2 input signal

thin layer of low refractive index cement Figure 2.5 Evanescent waves. (a) An evanescent wave exists in the medium of lower refractive index when total internal reflection occurs. (b) Frustrated total (internal) reflection across a narrow gap. (c) Fused fibre couplers (schematic); signal transfer is by frustrated total reflection. (d) Cubic beam splitter (schematic); one signal is generated by total internal reflection and one by frustrated total reflection 57 Colours Due to Refraction and Dispersion forbidden in terms of ray optics. The leaky portion of thewave decays rapidly and is called an evanescent5 wave, in contrast to the travelling, progressive or propagating waves described in Chapter 1. The existence of an evanescent wave is discovered if the reflected and transmitted wave amplitudes of the incident electromagnetic wave are determinedbysolutionoftheelectromagnetic wave equations for the appropriate boundary between two dielectric phases. The solution surprisingly shows that, when total internal reflection occurs, the waves not only have amplitude within the dielectric of high refractive index (corresponding to the total internal reflection) but also there is wave amplitude within the dielectric of low refractive index. This belongs to the evanescent wave. The evanescent wave is found to be a periodic wave, with the same period as the incident beam, but it decays exponentially on moving away from the interface between the dielectric phases. In general, the decay is quite abrupt and falls to negligible values within a couple of wavelengths, say approximately 1000 1500 nm (Figure 2.5a). Despite the fleeting nature of this wave, it has important properties that have been exploited in recent years. First, the evanescent wave can leak back into another dielectric as long as it is close enough to be within range of the rapidly decaying amplitude. Once inside this phase it will generate a propagating wave similar to the one that originally produced the evanescent wave itself. In the case where the two dielectric phases have the same refractive index, the new propagating wave is parallel to the original (Figure 2.5b). In this sense the original propagating wave has jumped the gap between the two dielectric phases and so has escaped total internal reflection a phenomenon called frustrated total (internal) reflection. This is of value in several areas of photonics, especially optical data processing, as it allows information carried by a laser beam to be introduced (coupled) into an optical fibre. There are a number of ways of building such a coupler. The simplest, conceptually, is simply to place a short section of the information-carrying fibre in close proximity to a section of the receiving fibre a fused-fibre coupler (Figure 2.5c). The evanescent wave transfers the data from the source to the receiver. This technique can also be used as a nondestructive sensor for signals in a fibre-optic waveguide. An information stream can also be divided into two streams with a beam splitter (Figure 2.5d). In this technology (as an example), a cube of glass is divided along a diagonal and the two halves are cemented together with a clear cement of low refractive index. Part of the signal beam is totally internally reflected at the diagonal prism face while part is transferred across the cement gap into the second prism. This phenomenon is not just of research interest. Coupling using evanescent radio-frequency waves lies at the heart of contactless charging of electronic devices, including batteries in heart pacemakers. Evanescent waves have another important property. Waves, in general, carry information. A normal light wave only carries information at a scale equal to or greater than the wavelength of the light itself. This imposes severe restrictions on the performance of optical instruments and essentially means that the resolution of an optical instrument, such as a microscope, is never more than the wavelength of the light used (Section 6.4). It is one reason why, in order to pack more electronic components onto silicon chips, lithographic techniques are continuously trying to use shorter and shorter wavelength radiation. Similarly, DVDs have a higher density of information content because they use shorter-wavelength red light, whereas CDs use longer wavelength infrared light. Evanescent waves, it turns out, carry information at a scale less than the wavelength of the light involved. This means that, if such waves can be used to form an image, the resolution will be less than the wavelength of light. Such lenses have now been fabricated with resolution a fraction of the wavelength of light (Section 2.10).

5 ‘Evanescent’ means ‘that which quickly passes away’ (OED). Colour and the Optical Properties of Materials 58

2.3 Refractive Index and Polarisability

To understand the relationship between refractive index and the atomic or molecular structure of a material it is necessary to recall that light can be treated as a varying electric field. If a static electric field is applied to an insulating material, the internal components which carry a charge will try to line up with the field and the material is said to become polarised. This means that any positively and negatively charged species present are rearranged slightly, with the positive charges moving in the field direction and the negative changes against it. Hence, some parts of the material take on a slightly positive charge while an equal number of parts become negative. The extent of this separation is measured as the relative permittivity (formerly dielectric constant)of the substance. The magnitude of the relative permittivity is found to be closely related to that of the refractive index. The most important of the internal components that contribute to the relative permittivity are (i) the permanent molecular dipoles present, (ii) the positive and negative ions present and (iii) the electrons present. In a static electric field, existing molecular dipoles will reorient themselves in the field as much as the surroundings will allow (Figure 2.6a). Similarly, a static electric field will cause the ions to move slightly so as to produce a net dipole moment (Figure 2.6b). The lightest component, the negatively charged electron cloud surrounding the atomic nucleus, is also deformed by an external field to create a dipole (Figure 2.6c). If the electric field is not static, but consists of an alternating field, the dipoles, ions and electrons will try to follow the changes in the field direction and move to and fro. (This effect is utilized in microwave ovens, which bombard the contents with radiation at frequencies of about 1010 Hz. As the applied electric field changes direction, the dipoles, especially those associated with water molecules, reorient to and fro. This continuous motion heats the food in the oven.) Motion is restricted for molecular dipoles, and when the frequency of the applied electric field becomes much higher than that of microwaves (of the order of 1011 Hz) any contribution of the dipoles is lost as the electric field is now changing too rapidly for them to keep up. The magnitude of the polarisability thus falls to a lower plateau (Figure 2.7). When the frequency of the field reaches that of near-infrared radiation (approximately 1014 Hz) even the lightest ions can no longer move to and fro quickly enough and their contribution to the polarisability is now lost. The magnitude of the polarisability then falls to a lower plateau (Figure 2.7). The electrons, however, can follow the oscillations of a varying electrical field even at visible and ultraviolet frequencies, and it is these which are most important in colour production. This response of the electrons to an applied alternating electric field is called the electronic polarisability. Classical electromagnetic theory considers that the oscillating electrons which are driven by the electric field of the light wave emit, in turn, an electromagnetic wave of the same frequency but at a reduced velocity compared with that in a vacuum, the difference being measured as the refractive index. (This, though, is true only for relatively low field strengths; see Chapter 3.) Theory relates the relative permittivity to the refractive index thus:

2 n ¼ er ð2:8Þ

where er is the relative permittivity of the material and n is the refractive index. Strictly speaking, Equation 2.8 applies when the relative permittivity is measured at optical frequencies. Recorded values of relative permittivity are often measured at frequencies far from those appropriate for light waves, and so the other contributions to the relative permittivity may be important. In such cases, the relationship given 59 Colours Due to Refraction and Dispersion

E (a) E=0

__ _ ++_ + +

__ __ (b) ++ ++

Figure 2.6 The effects of an external electric field E on the components of a solid. (a) Molecules with permanent dipoles (such as water, H2O) will align in the field as much as the structure will allow. (b) Ions which are evenly spaced (such as in rocksalt, NaCl) tend to be displaced in the field so as to create a net dipole moment. (c). A uniform electron cloud around an atom or an ion (such as lead, Pb) tends to distort so as to produce a dipole. The displacements have all been grossly exaggerated for clarity. Dipoles are indicated as arrows with the arrowhead at the positive end

in Equation 2.8 does not hold well especially for solids. A more useful relationship is given by the Lorentz Lorenz equation:

2 a n 1 ¼ N e ð : Þ 2 2 9 n þ 2 3eo

where N is the number of polarisable units in the material and ae is the electronic polarisability of each (identical) unit. This equation is only applicable to homogeneous isotropic materials that do not contain permanent dipoles or dipolar molecules. However, it is often taken to be approximately true for crystals of low Colour and the Optical Properties of Materials 60

dipoles + ions + electrons

dipolar contr bution

ions + electrons

ionic contribution

electrons Polarizability far mid near microwave infrared electronic visible contribution

1091011 10 10 1012 10 13 10 14 10 15 10 16 10 17 10 18 Frequency / Hz Figure 2.7 A simplified schematic illustration of the contribution of permanent dipoles, ions and electrons to the total polarisability of a material as the frequency of the applied field is increased. The contribution due to permanent molecular dipoles is lost when the field frequency reaches the microwave region and the contribution of the ions is lost at near-infrared frequencies. Only the effect of electronic polarisability occurs at optical frequencies symmetry, provided that they do not contain permanent dipolar molecules. In this case, because the different constituents will show different polarisabilities, these terms must be summed over the i different units to get the appropriate refractive index thus: 2 X a n 1 ¼ Ni i 2 n þ 2 3eo

In general, strongly bound electrons, trapped at atomic nuclei or in strong chemical bonds, have a low electronic polarisability and this leads to a low refractive index. Loosely bound electrons, such as outer electrons on large atoms or lone pair electrons, are highly polarisable and so will yield materials with a larger refractive index. This effect is well known. For example, lead oxide, PbO, contains large ions with a highly polarisable lone þ þ pair on each Pb2 ion. When lead oxide is added to ordinary glass the highly polarisable Pb2 ions (which occupy positions between the chains of SiO4 tetrahedra making up the structure) have the effect of considerably increasing the refractive index of the glass. Flint glass (which contains significant amounts of lead oxide), therefore, is prized and used as ‘cut glass’ and lead ‘crystal’ because the higher refractive index gives a more attractive appearance to the articles. Table 2.1 shows the effect of added PbO on the refractive index of three different flint glasses. Equation 2.9 is valid for most ordinary electric field strengths, and that applies to the electric field component of light. However, at high field strengths, such as those generated by intense laser light, it may not hold. In this case the refractive index of the material will be changed by the field. Solids in which this change can be made permanent are called photorefractive materials.

2.4 Refractive Index and Density

As the previous section demonstrated, electronic polarisability and the number of polarisable atoms present are important factors that contribute to the magnitude of the refractive index. Gases, because of their low densities, 61 Colours Due to Refraction and Dispersion

Table 2.1 Refractive indices

a a Substance Refractive index n Substance Refractive index n Vacuum 1.0 (definition) Dry air, 1 atm 15 C 1.000 27 b Water 1.3324 Na3AlF6 (cryolite) 1.338 b MgF2 1.382 Fused silica (SiO2) 1.460 KCl (sylvite) 1.490 Crown glass 1.522 c Extra light flint glass 1.543 NaCl (halite) 1.544 c Flint glass 1.607 MgO (periclase) 1.735 c b Dense flint glass 1.746 Al2O3 (corundum) 1.765 b ZrO2 (zirconia) 2.160 C (diamond) 2.418 b CaTiO3 (perovskite) 2.740 TiO2 (rutile) 2.755 a A value appropriate to the yellow light emitted by sodium atoms (the sodium D lines; Chapter 7), with an average wavelength 589.3 nm, is given. b The refractive index varies with direction; the average value is given. c The flint glasses contain significant amounts of lead oxide, PbO, as follows: extra light flint, 24 mass% PbO; flint, 44 mass% PbO; dense flint, 62 mass% PbO. have refractiveindices close to unity. However, although small, thevariation of the density of air as a function of temperature is the source of mirages and related visual effects (Section 2.5). Densely packed arrays of atoms in liquids and solids have a higher refractive index than gases. The refractive index increases with density, as can be confirmed from the Lorentz Lorenz equation, (Equation 2.9). The number of scattering centres per unit volume can be expressed as a density, so that the equation can be written:

n21 ¼ rr n2 þ 2 s where n is the refractive index, r is the density measured at the same temperature as the refractive index and rs is a constant the speciﬁc refraction. The molar refraction Rm of a compound is deﬁned as rs times the molar mass, so that:

ðn21ÞV m ¼ R n2 þ 2 m

Although the refractive indices of most simple compounds are known, it is sometimes useful to estimate the refractive index of more complex or hypothetical materials. One of the most successful ways of doing this is via the Gladstone Dale equation, which combines density and, indirectly, polarisability terms. It is especially useful for complex oxides, for which the Gladstone Dale formula can be written:

n ¼ 1 þ rð p1kr1 þ p2kr2 þ p3kr3 þ Þ or X n ¼ 1 þ r pikri ð2:10Þ where r is the density of the complex oxide and the terms pi and kri are deﬁned below. The assumption underlying the formula is that the refractive index of a complex oxide is made up by adding together the contributions from a collection of simple oxides, oxide 1, oxide 2 and so on, for which optical data are known. The polarisability is taken into account by allocating to each of the simple oxide components a factor kr called the refractive coefﬁcient, an empirically determined constant. The amount of each oxide is taken into account Colour and the Optical Properties of Materials 62

by multiplying the refractive coefﬁcient by its weight fraction p in the compound. A number of values of kr for use in the Gladstone Dale formula are given in Table 2.2. The rule works well and usually gives answers within about 5 %. Note, however, that the value obtained is an average refractive index. Many oxides have refractive indices which vary according to crystallographic direction. The Gladstone Dale relationship ignores this feature. The equation can also be used to determine a value of either density or average refractive index for unknown polymorphs of simple oxides. For example, the Gladstone Dale equation for the polymorphs of SiO2 is:

n ¼ 1 þ 0:208r and for the polymorphs of TiO2 it is:

n ¼ 1 þ 0:393r

Provided the density of each polymorph is known, its average refractive index can be found and vice versa.

2.5 Invisible Animals, GRINs and Mirages

As mentioned earlier (Section 1.16), the initial premise of the H.G. Wells tale The Invisible Man is that if the refractive index of the body matches that of air, the body would become invisible, and this principle is widely used by aquatic animals to avoid predators. These creatures, typified by jellyfish, have a gelatinous body which has a refractive index very close to that of water. This renders them more or less invisible. Indeed, some creatures succeed so well at this that they cannot be detected until the observer is only centimetres away. The relationship between density and refractive index can be exploited quite simply to make materials with a lower than normal refractive index. A practical use of this idea, conceived more than 50 years ago, is to fabricate the material in the form of foam. Provided that the air bubbles are smaller than the wavelength of light they are not resolved and the light encounters a medium in which the effective refractive index lies between that of air and that of the foam matrix. For example, silica containing pores of the about 4 nm in diameter has been fabricated with refractive index of 1.23, compared with the refractive index of a nonporous film, 1.457. Such structures are used in antireflection coatings (Section 3.7). The refractive index of porous materials depends upon the pore shape and distribution, as well as the phase that fills the pore. The polarisation and wavelength of the light are also important variables. To a first approximation, the refractive index of the whole, nt, can be assessed as that of a simple mixture:

nt ¼ nmVm þ npVp ð2:11Þ where nm is the refractive index of the material that can be regarded as the matrix, Vm is the volume fraction of the matrix, np is the refractive index of the material ﬁlling the pore and Vp is the volume fraction of the pore phase. The volume fractions are given by:

volume of matrix phase V ¼ m total volume and

volume of pore material V ¼ p total volume 63 oor u oRfato n Dispersion and Refraction to Due Colours

Table 2.2 Refractive coefﬁcients for some oxidesa

b Oxide kr Oxide kr Oxide kr Oxide kr Oxide kr Oxide kr Oxide kr Oxide kr Oxide kr

H2O 0.340 Li2O 0.307 BeO 0.240 B2O3 0.215 CO2 0.221 N2O5 0.242 Na2O 0.190 MgO 0.200 Al2O3 0.207 SiO2 0.208 P2O5 0.183 K2O 0.196 CaO 0.210 TiO2 0.393 Ga2O3 0.170 GeO2 0.167 As2O5 0.162 Rb2O 0.128 SrO 0.145 Y2O3 0.170 ZrO2 0.211 Nb2O5 0.268 MoO3 0.237 In2O3 0.130 SnO2 0.143 Sb2O5 0.153 Cs2O 0.119 BaO 0.128 La2O3 0.148 Ta2O5 0.151 WO3 0.171 PbO 0.133 Bi2O3 0.139 a Data from: J. A. Mandarino, Can. Mineral., 14, 498–502 (1976); 16, 19–174 (1978); 17, 71–76 (1979); 19, 441–450 (1981). b 3 3 3 These values give correct results if the density is in g cm . To use density in kg m , multiply the values of kr by 10 . Colour and the Optical Properties of Materials 64 that is:

Vm þ Vp ¼ 1

If there are several different types of pore material present then the equations can be extended; for example:

nt ¼ nmVm þ np1Vp1 þ np2Vp2 þ where

Vm þ Vp1 þ Vp2 þ ¼1

The equations can be written in terms of the weight fraction of the components W1, etc. by substituting via the density equation:

¼ W1 V1 r 1

Although the refractive index of most isotropic materials is uniform, transparent solids with a varying refractive index are manufactured for a number of purposes. These are called graded-index (GRIN) materials. AGRIN solid can be made by arranging a nonuniform distribution of dopants throughout the bulk. Thus, GRIN optical ﬁbres (Section 2.9) are made by the diffusion of GeO2 into SiO2. The optical path of a ray in a GRIN material will follow a curve, the form of which depends upon the refractive index distribution in the material (Figure 2.8a). Cylinders of materials with an appropriate refractive index variation can then act as a focusing lens (Figure 2.8b). The optical path length [d] in a material with a smoothly varying refractive index is obtained by replacing the summation sign in Equation 2.6 by an integral, which for a path between points a and b is:

ðb

½d¼ nsð pÞ dp a where ns( p) is the refractive index in the substance at position p and dp is a small element of path (Figure 2.8c). GRIN solids are not uncommon in nature. The lens of the human eye is an example. It is built up of layers with a refractive index which varies from about 1.41 at the centre to 1.39 at the outer layers. The atmosphere also has a continuously varying refractiveindex, from 1.0 in space to approximately 1.000 292 for dry air and light of wavelength 589.3 nm (the mean of the sodium D lines, Chapter 7) at 0 C and 1 atm pressure. The refractive index will vary with pressure, temperature and the content of other gases, especially water vapour. In particular, local density fluctuations can have a considerable effect on the refractive index and give rise to a variety of meteorological phenomena, such as mirages. In general, a temperature gradient in the air changes the refractive indexof the air and sets up an ‘air lens’. Because the lens is imperfect, the images reaching the eyeare imprecise. For this reason, the human imagination has constructed a variety of fanciful explanations for the apparitions observed, including the familiar water pools and more arcane Atlantis myths. The idea of using GRIN optics was evolved in some night-flying insects some millions of years ago. A number of animals, notably moths, have eyes well adapted to night vision. The surface structure of these ‘moth eyes’ is bumpy and acts like a GRIN layer with a refractive index between that of the surrounding medium and the substrate. The net result is to cut down or eliminate surface reflection (see Section 3.7.3). 65 Colours Due to Refraction and Dispersion

(a)

(b)

n Figure 2.8 GRIN materials. (a) The path of a ray in a GRIN is generally curved. (b) Suitable refractive index variation can produce a focusing effect. (c) The path length in a GRIN material is speciﬁed by the integral of the refractive index ns(p) at point p over the length of the path dp

2.6 Dispersion and Colours Produced by Dispersion

As mentioned above, the refractive index is far from constant. The variation of the refractive index of a transparent material with wavelength is known as dispersion (Figure 2.9). Dispersion can be formally deﬁned as the slope of the refractive index n versus the wavelength l curve, dn/dl. In general, the index of refraction increases as the wavelength decreases, so that the refractive index of red light in a material is less than that of violet light. This situation is referred to as normal dispersion. Although the normal dispersion of many materials is rather small, it is important to include it when calculating the optical properties of lenses and similar high-quality optical components. Anomalous dispersion is found in the region of absorption bands in the material, when transparency is lost. These absorption bands are associated with transitions from one energy conﬁguration (often the ground state) to higher energy levels. For many transparent materials a good representation of the variation of refractive index with wavelength in the visible region is given by Cauchy’s equation:

B C n ¼ A þ þ l2 l4 Colour and the Optical Properties of Materials 66

(a)

normal dispersion

anomalous dispersion Refractive index

absorption band Wavelength

1.50 1.85 aluminium oxide (corundum, sapphire)

fused silica glass Refractive index Refractive index

1.45 1.80

400 500 600 700 400 500 600 700 Wavelength / nm Wavelength / nm (b) (c) Figure 2.9 The variation of refractive index with wavelength. (a) Schematic dispersion curve for a transparent material. Anomalous dispersion occurs close to energy transitions from a lower to a higher energy level. (b) Dispersion curve for fused silica glass. (c) Dispersion curve for corundum, Al2O3. In the case of corundum the refractive index depends upon direction and average values are plotted

where A, B and C are empirically determined parameters. For lens design Cauchy’s equation is not sufﬁciently precise, and a more accurate formula, which gives the refractive index of glasses in the wavelength range 365 2300 nm to high degree of ﬁdelity, is the Sellmeier equation:

= l2 l2 l2 1 2 ¼ þ B1 þ B2 þ B3 n 1 2 2 2 l C1 l C2 l C3

where the wavelength l is in micrometres and B1 B3 and C1 C3 are the Sellmeier constants appropriate to the glass. The Sellmeier equation can also be applied to transparent crystals. For those that are not isotropic, different equations must be obtained for each of the crystallographically independent directions. 67 Colours Due to Refraction and Dispersion

The Abbe V-value,orAbbe number, written Vd, awidely used measure of the dispersion of a transparent solid, is given by:

nd 1 Vd ¼ nF nC where nd is the refractive index of the material at a (yellow) wavelength, 587.56 nm (the helium d-line; see Chapter 7), nF is the refractiveindex of the material at a (blue) wavelength of 486.1 nm (the hydrogen F line) and nC is the refractive index of the material at a (red) wavelength of 656.3 nm (the hydrogen C line). The reciprocal of the Abbe number is often called the dispersive power. The dispersion of refractive index is the cause of the formation of a spectrum when white light is passed through a glass prism (Figure 2.10a). Snel’s law tells us that, for a given angle of incidence i, sin r is inversely

(a)

θ i θ r

white light silica glass prism

(b) white red

violet

white white

violet

lens red white

(c)

α white δ red violet

Figure 2.10 (a) The refraction of white light by a silica glass prism. For silica glass, the refractive index for violet light is greater than for red light, which disperses the light to form a spectrum. (b) The edge of a simple lens acts as a prism and so causes chromatic aberration. (c) The deviation of light by a thin prism provides a useful model for the dispersion from a thin lens. Each colour will be deviated by a different amount d ¼ (nl 1)a Colour and the Optical Properties of Materials 68

white violet red

cut diamond

Figure 2.11 The combination of high refractive index and high dispersion in cut diamonds gives these gemstones the ability to produce spectral colours, known as ﬁre

proportional to the refractiveindexn, so that as n increases r decreases and the ray deviates more. Red light then tends to be the least deviated and violet light the most. The higher the dispersion, thewider will be the spectrum. Exactly the same effect is found in simple lenses. The edge of the lens is approximately prism shaped and dispersion causes the image to become coloured at the periphery of the ﬁeld of view, (Figure 2.10b). This effect is known as chromatic aberration. If the lens is considered to be a thin prism, the angle of deviation of the rays d will be given by:

d ¼ðnl1Þa where nl is the refractive index appropriate to the colour and a is the small angle at the top of the prism (measured in radians) (Figure 2.10c). Chromatic aberration is avoided in expensive lenses by using combinations of glasses chosen so as to compensate for the effects of dispersion in each component. Such compound lenses are called achromats. Dispersion is responsible for the flashes of colour, called fire, that are such an important feature of diamonds. The production of such fine colours is due to the combination of very high refractive index and high dispersion. The stones are ‘cut’ (actually cleaved) so as to produce many facets, each of which can act as a tiny prism, thus greatly enhancing the display of fire as the gem moves (Figure 2.11).

2.7 Rainbows

The rainbow is one of the most beautiful examples of colour produced by refraction (Figure 2.12). Most frequently seen, when the observer’s back is to the sun, is a single bright arc called the primary rainbow (Figure 2.13a). The colour violet is always innermost, at an angle of 41 to the incident beam. The colours proceed through indigo, blue, green, yellow, orange to red on the outside of the arc at an angle of 43 to the incident beam. The locus of the various angles generates the arc seen and the observer appears to be at the apex of a cone with an average semi-vertical angle of about 42 (Figure 2.13b). A careful examination of the sky near a rainbow will often, but not always, show many other features, including a fainter secondary rainbow at an angle of about 50 and various supernumerary arcs inside the primary bow. The secondary bow is higher in the skythan the primary bow, is much less intense than the primary and the colour sequence is reversed with respect to the primary bow. Also, though not so easily seen, is the fact 69 Colours Due to Refraction and Dispersion

Figure 2.12 Primary and secondary rainbows. The secondary bow is higher in the sky than the primary bow and is noticeably fainter. The colour sequence in the secondary bow is reversed compared with that in the much brighter primary bow. The region of sky between the two bows is noticeably darker than the sky above and below the two bows. This is Alexander’s dark band that the sky between the two bows is noticeably darker than the sky below or above the bows. This darker region is known as Alexander’s dark band (Figure 2.14). Although a complete description of all of these features is complex, the (first-order) explanation of the primary rainbow is relatively simple.6 It is produced by a single reflection from inside a raindrop (Figure 2.15a). The point where the incident light beam falls on the drop can be defined by the impact parameter IP, which lies between 0 and 1.0 and is expressed as a fraction of the drop radius R (Figure 2.15b). From the geometry of the refraction and internal reflection it is seen that the deviation d of a ray (Figure 2.15c) is given by:

d ¼ 180 þ 2i4r sin i ¼ IP IP sin r ¼ n where i is the angle of incidence, r is the angle of refraction and n is the refractive index of water. It is possible to calculate values of d as a function of IP (Table 2.3).

6 The discussion here is termed ‘ﬁrst order’ because raindrops are not spherical as they fall through the air. They have a ﬂattened bottom and have a shape more like a ladybird than a ball. Colour and the Optical Properties of Materials 70

(a) primary rain rainbow

sunlight red

violet

43 41 observer

(b)

raindrop sunlight ~138°

rainbow

Figure 2.13 The geometry of a primary rainbow. (a) To observe a rainbow, sunlight must come from behind the observer. (b) In the main or primary bow, each raindrop generates a cone of refracted and reﬂected light; the violet light appears to come from a cone of semi-vertical angle 41 and the red light from a cone of semi-vertical angle 43

It is seen that the deviation falls gradually and reaches a minimum at about 138. In fact, there is a considerable bunching of the rays in the region of this minimum deviation, which is the reason why the rainbow appears to be bright. The exact minimum can be determined analytically by the differentiation of the formula for the deviation of the ray. The result is found to be:

IP ¼ 0:862 38; d ¼ 137:6 where the refractive index of water has been taken to be 1.33. The refractive index of natural water depends upon any dissolved impurities, the temperature and the wavelength of the light. For accurate results, precise values for the refractive index of water are required. Reasonable values to take are:

n ðred; 20 CÞ 1:3310; n ðviolet; 20 CÞ 1:3440 71 Colours Due to Refraction and Dispersion

violet secondary rain rainbow red

Alexander’s Dark Band

sunlight red primary rainbow violet

129°

138°

51 42 observer Figure 2.14 The positions of the primary and secondary rainbows. The region between them is Alexander’s dark band

Using these, it is found that the minimum deviation of red light is 137.63 and violet light is 139.35, giving an angular width of 1.72 for the bow. As the different colours diverge on leaving a drop, they do not all enter the eye. In fact, each colour observed comes from a different raindrop, appropriately positioned with respect to the observer. Normally, all of a drop is illuminated in sunlight and so every impact parameter occurs over every angle in the vertical plane, so that, as described above, the refracted and reflected light is deviated into a cone with a semi- vertical angle of approximately 42 and with most intensity concentrated into the outer surface. This means that each of the colours that enters the eye from a rainbow originates in a separate arc of water drops. For the same reason, no two observers ever see exactly the same rainbow. Each person sees only a unique part of the raindrop curtain that subtends the correct angles with respect to the observing eye. The secondary bow is caused by two internal reflections (Figure 2.16a and b). The geometry drawn in this figure allows one to conclude that:

d ¼ 360 þ 2i6r sin i ¼ IP IP sin r ¼ n Colour and the Optical Properties of Materials 72

(a) sunlight

43° water 41° drop

violet red observer

(b) i r IP r i r

r i

Figure 2.15 The reflection and refraction that a ray of light undergoes in forming a primary rainbow. (a) The primary bow is produced by a single reflection within each raindrop combined with dispersion of the light due to the variation of refractive index with wavelength. (b) The refraction and reflection within a rain drop. (c) The angle of deviation of the reflected ray

where d is the deviation of a ray, i is the angle of incidence, r is the angle of refraction, IP is the impact parameter and n is the refractive index of water (Table 2.4). To compare this with the primary bow, the incident ray to be considered needs to enter the drop below the centre line. The relevant angle of deviation for comparison with that of the primary bow is (360 d ) (Figure 2.16c and d). Table 2.4 shows that the deviated rays cluster, this time with a minimum deviation of close to 230 (or a maximum value of (360 d ) of 129). The angle that the bow subtends to the observer is close to 51, which indicates that the secondary bow will lie above the primary bow (Figures 2.12 and 2.13). 73 Colours Due to Refraction and Dispersion

Table 2.3 The deviation of a ray which forms a primary rainbowa

IP i/deg r/deg d/deg 0.1 5.74 4.31 174.2 0.2 11.54 8.65 168.5 0.3 17.46 13.04 162.8 0.4 23.58 17.50 157.2 0.5 30.00 22.08 151.7 0.6 36.87 26.82 146.5 0.7 44.43 31.76 141.8 0.8 53.13 37.00 138.3 0.9 64.16 42.59 138.0 0.95 71.81 45.58 141.3

a Calculated using n ¼ 1.33.

(a) i r

IP r r

i r

r r

(b)

Figure 2.16 The reflection and refraction that a ray of light undergoes in forming a secondary rainbow. (a) The secondary bow is produced by a double reflection within each raindrop combined with dispersion of the light due to the variation of refractive index with wavelength. (b) The angle of deviation of the reflected ray. (c) As (a) with an impact parameter below the centre of the drop. (d) The angle of deviation of the reflected ray in (c) Colour and the Optical Properties of Materials 74

(c)

r r

r i

r r IP r i

(d) 360 - d

Figure 2.16 (Continued)

Table 2.4 The deviation of a ray which forms a secondary rainbowa

IP i/deg r/deg d/deg (360 d )/deg 0.1 5.74 4.31 345.6 14.3 0.2 11.54 8.65 331.2 28.2 0.3 17.46 13.04 316.7 43.3 0.4 23.58 17.50 302.2 57.8 0.5 30.00 22.08 287.5 72.5 0.6 36.87 26.82 272.8 87.2 0.7 44.43 31.76 258.3 101.7 0.8 53.13 37.00 244.2 115.7 0.9 64.16 42.59 232.8 127.2 0.925 67.67 44.07 230.9 129.1 0.95 71.81 45.58 230.1 129.9 0.975 77.16 47.15 231.4 128.6 a Refractive index of water taken as 1.33. 75 Colours Due to Refraction and Dispersion

As in the case of the primary bow, the value of the maximum deviation can be determined precisely be differentiation. The result is found to be:

IP ¼ 0:950 73; 360 d ¼ 129:9 where the refractive index of water has been taken to be 1.33. Using more precise values for the refractive index of water, we find that the maximum deviation of red light is 129.63 and for violet it is 126.52, giving an angular width of 3.11 for the bow. It also confirms that the colours in the secondary bow are reversed with respect to the primary bow. In addition, because a largeamount of light is not reflected internally, each additional internal reflection decreases the intensity of the bow considerably. A secondary bow is, therefore, rarely very intense, being only about 43 % as bright as the primary. The calculations shown in the tables indicate that light rays have a minimum deviation from 180 to about 138 for the primary bow and a maximum deviation from 0 to about 129 for the secondary bow. No light enters the space lying between 129 and 138, the region between the primary and secondary bows. Observation will show that this region is indeed darker than the sky below the primary bow and above the secondary bow. It is referred to as Alexander’s dark band (Figures 2.12 and 2.13) Other rainbows are produced by more internal reflections within the water drops. Although higher order rainbows are too faint to see in the sky, they can be observed in the laboratory. Three internal reflections produce a ternary bow and so on, and up to a dozen orders can be seen indoors. The formula for the deviation d of a light ray after m internal reflections, generating the mth order rainbow, is:

d ¼ 2ði rÞþmð180 2rÞ

The maximum or minimum deviation of the rays is given by:

½ðm þ 1Þ2 1ðIPÞ2 ¼ðm þ 1Þ2 n2 where m is the order of the bow, n is the refractive index of water and IP is the impact parameter. Rainbows are found to be polarised even though sunlight, the cause of the bow, is unpolarised. This is explained in Chapter 3.

2.8 Halos

A halo is a rather pale diffuse ring of colour, often red on the inner side and blue on the outer side, seen around a bright object such as the sun or moon when partially obscured by thin, high cloud (Figure 2.17a). Halos are not as spectacular as rainbows and can often be missed during casual observation. The total angular width of a halo is 44. Once again, refraction and dispersion are responsible for the colour, but in this instance the refraction occurs within randomly oriented hexagonal ice crystals in the upper atmosphere. The commonest halo, the 22 halo, is a ring subtending an angle of 22 to the observer’s eye (Figure 2.17b). Dispersion of the blue wavelengths is more than that of the reds, and so the halo is red internally and blue violet externally. Figure 2.18 shows a halo-like arc of colour due to refraction and dispersion in ice crystals in stratospheric clouds.

2.9 Fibre Optics

2.9.1 Optical communications

In the early years of the twentieth century, data transmission was mainly by way of electrical impulses sent along copper wires, by radio waves or by manual transport of written records. In the early years of the Colour and the Optical Properties of Materials 76

(a)

sun violet red

(b)

ice crystal

60°

~22° white light red violet

Figure 2.17 (a) A halo is sometimes seen around a bright object such as the sun when partially obscured by high cloud. The total angular width of a halo is 44. (b) The halo is formed by refraction of light through a random array of hexagonal ice crystals. The average deviation of the light in each crystal is 22, with the deviation of the red ray about 1.5 less than that of the blue (The angles are exaggerated for clarity)

Figure 2.18 A halo-like arc of colour produced by reflection and dispersion when sunlight falls upon prismatic ice crystals in the upper atmosphere 77 Colours Due to Refraction and Dispersion twenty-first century, optical data transmission along glass fibres is normal. Tyndall, in 1870, first showed that light could be transmitted along a jet of water even if the path was curved. The reason for the transmission is that total internal reflection at the water air interface prevents the light from escaping. Shortly after this, the transmission of light within a glass rod was also demonstrated. As glass can be easily dawn into fine fibres it soon became clear that bundles of fibres could be used for the remote illumination and viewing of inaccessible or dangerous areas. The subject of light transmission along thin fibres of glass, plastic or other transparent materials is referred to as fibre optics. Until the mid 1950s fibre optics remained something of a curiosity. This was because the transparency of the glass was poor, due to the presence of impurities. Moreover, different colours tended to separate because of the dispersion of the refractive index of the glass, resulting in strong chromatic aberration and producing unsatisfactory images. Despite these problems, use was made of short lengths of glass fibres for decorative purposes and lighting. However, during these years the main use of glass fibre bundles was in medicine, making the examination of internal organs possible without surgery. The situation changed in the mid 1960s. The impetus was for rapid communication of large amounts of data along securelines,and forthis glass fibresweredeemedideal. Asa spin-offfromdevelopments incommunication technology, uses of fibre bundles as long-distance light guides, for remote viewing of inaccessible objects or dangerous devices, in medical imaging and in many applications of laser technology has also burgeoned. The developments which led to these changes are described, with emphasis on materials properties. Information upon engineering aspects will be found in the sources listed in this chapter’s Further Reading.

2.9.2 Optical ﬁbres

Data is carried in optical communications by a series of pulses of light encoded so that information can be stored and retrieved. The transparent optical wave carrier used for communications is silica (SiO2) glass. The light pulses launched into the fibre are constrained to stay within the fibre by total internal reflection. Thus, the core of the fibre, along which light travels, must possess a higher refractive index than the outer surface of the fibre. Moreover, a glass surface at which the total internal reflection is to occur is easily damaged and needs protection. Both of these objectives are met by providing a surface cladding of lower refractive index glass, compared with the core of the fibre. The core and the cladding make up a single glass fibre (Figure 2.19). The cladding should not be confused with a plastic protective covering, which has no optical role to play. The starting point for a fibre is a high-purity silica tube containing only a few parts per million of hydroxyl ions. In order to create a fibre with the correct refractive index profile, the silica tube is rotated and heated while a gas consisting of various amounts of silicon tetrachloride (SiCl4), germanium tetrachloride (GeCl4), phosphorus oxychloride (POCl2), Freon (a chlorofluorohydrocarbon, typified by CF2Cl2) and oxygen is

plastic cover

cladding n ~ 1.48 optical fibre core n ~ 1.5

Figure 2.19 The structure of a silica optical fibre. The core has a higher refractive index than the cladding, which serves to confine light rays by total internal reflection. The fibre is covered with a protective plastic coating Colour and the Optical Properties of Materials 78

reactants:

SiCl4 silica tube

GeCl4

POCl2 (Ge, P, F)1-xx (Si) O2 Freon O 2 rotate and heat (a)

collapse

cladding

core

(b) preform rod Figure 2.20 The formation of a preform rod for silica ﬁbre production. (a) Reactive gases are passed through the centre of a hot, rotating silica tube, where they decompose to form layers of (Ge,P,F)xSi1xO2. (b) After reaction, increased heating causes the tube to collapse to the perform with the cladding (lower dopant content) and core (higher dopant content) at the centre allowed to ﬂow through its centre (Figure 2.20a). At the temperatures of the tube, 700 900 C, the gases decompose:

SiCl4 þ O2 ! SiO2 þ 2Cl2

GeCl4 þ O2 ! GeO2 þ 2Cl2

4POCl2 þ 3O2 ! 2P2O5 þ 4Cl2

The result is that a ‘soot’ consisting of a mixture of silicon, germanium and phosphorus oxides forms on the inside of the tube. As the heating zone traverses the tube the ‘soot’ merges with the tube to form a glass inner coating about 10 mm thick. The refractive index variation is achieved in two stages. Of the order of 12 to 32 layers are deposited initially to form an inner coating inside the tube which will become the cladding on the fibre. This has a composition of SiO2 containing 0.1 to 1 % P, F and Ge, the fluorine incorporation arising from the Freon gas present. The refractive index is close to that of pure SiO2. Following this stage the material which will ultimately form the core is deposited. This is achieved by depositing 4 to 10 layers of material with an overall composition somewhere between the limits (Ge,P,F)0.06Si0.94O2 to (Ge,P,F)0.3Si0.7O2, depending upon final use. The replacement of Si by the heavier Ge and to a lesser extent P increases the refractive index over that of the first layers laid down. When sufficient layers have been formed the temperature is raised enough to cause the tube to collapse under surface tension. The result is a solid rod with a centre of higher refractive index glass surrounded by a region of lower refractive index glass enveloped in the original silica of the tube. This solid rod is called a preform (Figure 2.20b). To transform the preform into a fibre by a process called fibre drawing, the end of a preform rod is softened to near to its melting point. Under these conditions glass has the property that it can be pulled out and will form a long fibre. Surprisingly, the refractive index profile of the preform is preserved exactly in the fibre even though the preform diameter of 15 100 mm is drawn down to approximately 0.1 mm. 79 Colours Due to Refraction and Dispersion

2.9.3 Attenuation in glass ﬁbres

Attenuation describes the loss of light intensity as the signal is transmitted along the ﬁbre. This is of major concern, as any degradation of the signal must be minimized. The loss is deﬁned as:

ð Þ¼ ½ ð Þ= ð Þ loss dB 10log10 P x P 0 where P(0) is the power input, at x ¼ 0, and P(x) is the power at a remote point x. The attenuation is deﬁned as the loss per kilometre; thus:

10log ½PðxÞ=Pð0Þ attenuation ¼ 10 x The units of attenuation7 are decibels per kilometre. Ordinary window glass has an attenuation of about 100000 dB km 1. Attenuation, like dispersion, varies with wavelength. The spectral response of a fibre defines the way in which the fibre attenuation changes with the frequency of the radiation being transmitted. Attenuation is caused by a combination of absorption and scattering within the glass. Extrinsic attenuation is due to poor processing or fabrication techniques, and may be due to artefacts such as bubbles, particles, impurities and variable fibre dimensions. These problems have been eliminated in modern optical fibre manufacture. Intrinsic attenuation is a property of the pure material itself, and cannot be removed by processing. It is the ultimate limit on the performance of the fibre and mainly arises from two factors: Rayleigh scattering (Section 5.2) and lattice vibrations. Rayleigh scattering is due to small density fluctuations in the glass. This variation is an inevitable feature of the noncrystalline state and cannot be removed by processing. As Rayleigh scattering is proportional to l 4, where l is the wavelength of the optical pulse, the effect is more important for short-wavelength radiation. For anyparticularglass, most of the factors affecting Rayleigh scattering are constant and cannot be easily changed. However, materials with a low refractive index and glass transition temperature tend to exhibit low Rayleigh scattering. Absorption due to lattice vibrations, referred to as phonon absorption, occurs when the lattice vibrations of the solid match the energy of the radiation. This occurs for infrared wavelengths, and converts the signal energy into heat. It is a function of the mass of the atoms in the glass and the strength of the chemical bonds between them and results in a decrease in the transparency of the glass at long wavelengths. Absorption due to electronic transitions (Chapter 7), mostly at high energies and associated with ultraviolet wavelengths, does not figure significantly in present-day applications, but may become important if shorter signal wavelengths are to be used in the future. The dependence on wavelength of absorption due to electronic transitions can often be expressed by a formula of the type: ¼ B2 Electronic absorption B1exp l where B1 and B2 are constants relating to the glass used and l is the wavelength of the radiation. By 1979, the best silica fibres showed only intrinsic attenuation and had a loss of about 0.2 dB km 1 at 1.5 mm wavelength. The current industry standard is slightly less than this, at about 0.16 dB km 1. Despite this achievement, new fibre materials are constantly being explored. The absorption maxima caused by lattice vibrations can be manipulated both by changing the strength of the chemical bonds between the components and by changing the mass of the atoms linked. For example, silica, with rather light

7 The unit of loss is the decibel, dB; the base unit, the bel, is almost never used. Colour and the Optical Properties of Materials 80 atoms and strong bonds, transmits satisfactorily only to about 2.5 mm and strongly absorbs radiation in the 8 15 mm wavelength range. One group of materials with great promise are glasses primarily made from fluorides of zirconium (ZrF4), barium (BaF2) and lanthanum (LaF3), called ZBLAN glasses. Although the chemical bonding in fluorides is regarded as being as strong as in oxides, the heavy atoms move the absorption maximum to wavelengths of 17 25 mm. In addition, they are found to have an attenuation which is only one-hundredth of that of silica. They are, therefore, enormously attractive for long-distance high data density communications. Desirable characteristics are also found in the arsenic triselenide (AsSe3) glasses, composed of very heavy atoms and linked by weak bonds. These do not absorb strongly until 44 46 mm and so show great potential for the transmission of infrared radiation. Unfortunately, the chemical difficulties associated with making fluoride and selenide glasses have not yet been solved and they are not currently used in long-distance commercial applications.

2.9.4 Chemical impurities

The preparation of high-purity glass was one of the most important advances needed to allow fibre-optic communications to become a reality, and enormous strides in improvement of glass purity have been made since the earliest times (Figure 2.21). In the original glass fibres, transition metal impurities caused difficulties because they absorb strongly in the visible. The gravest problem was iron, present as Fe2 þ , and it is this ion which gives window glass its greenish tint (Section 7.7). Even as low a concentration as 1 ppm of iron can result

107

6 10 ordinary “window glass”

105 high purity melting high quality lenses 104

103 highest quality optical glass

chemical vapour deposition

Attenuation / (dB km) 100 optical fibres drying 10

1000 500 800 1200 1600 1800 1900 1960 1970 1980 1990 BC BC AD Year Figure 2.21 The quality of glass through the ages. Recent improvements have been in response to the needs of optical ﬁbre manufacturers. Currently, silica ﬁbres are routinely made with an attenuation of less than 0.2 dB km 1 81 Colours Due to Refraction and Dispersion -1

10.0 -1

1.0

Attenuation / [dB km ppm (OH-) ] 0.10

0.01 700 800 900 1000 1100 1200 1300 1400 Wavelength / nm Figure 2.22 The attenuation introduced into a silica fibre by the presence of OH in an attenuation of 15 dB km 1. The presence of transition metal cations was overcome by the preparation of silica using veryhigh purity chemicals made available by the semiconductor industry. At present it is possible to purchase silica with no significant transition metal ion impurities present. The most important impurity in silica fibres today remains hydroxyl ( OH) (Figure 2.22). Hydroxyl arises from water or hydrogen incorporation into the glass during fabrication. Flames used to melt silica are rich in both of these impurities, and any silica melted in a gas flame will be heavily contaminated by hydroxyl. The main absorption peaks are at 950, 1240 and 1390 nm and an impurity OH level of 1 ppm can give an attenuation of the order of 102 dB km 1 at 1.4 mm signal wavelength. It is clear, therefore, that silica for optical fibre use must be melted in electric furnaces in a dry atmosphere to eliminate hydroxyl as much as possible. Despite careful processing, fibres currently in production still contain significant amounts of hydroxyl, which remains an important source of attenuation. When hydroxyl absorption is superimposed upon the intrinsic attenuation of a pure silica glass, of which Rayleigh scattering is the main contributor at shorter wavelengths and phonon absorption (infrared absorption) at longer wavelengths, it is seen that the best window for signals is close to 1500 nm (Figure 2.23).

2.9.5 Dispersion and optical-ﬁbre design

A short pulse of light launched into a fibre will tend to spread out, due to dispersion (Figure 2.24a c). When dispersion was discussed above it was defined in terms of the change of refractive index with wavelength. In optical fibres, the dispersion is defined as the delay between the arrival time of the start of a light pulse and its finish time relative to that of the initial pulse. It is measured at half peak amplitude (Figure 2.24d). If the initial pulse has a spread of t0 seconds at 50 % amplitude and the final pulse a spread of tx seconds at 50 % amplitude after having travelled x kilometres, the dispersion is given by: t t dispersion ¼ x 0 x The units of dispersion in optical fibres are nanoseconds per kilometre. Colour and the Optical Properties of Materials 82

intrinsic loss 1.0 hydroxyl absorption -1 0.5

Loss / dB km 0.3 Rayleigh silica infrared scattering absorption

0.2

0.1 1.1 1.2 1.3 1.4 1.5 1.7 2.0 Wavelength / μm Figure 2.23 The attenuation in a silica ﬁbre due to intrinsic attenuation and OH impurities

Dispersion will obviously arise if the light used is not monochromatic. An initially sharp pulse consisting of a group of wavelengths will spread out as it travels down the fibre because the refractive index depends on wavelength, and thus the light of different wavelengths will travel at different speeds. This effect is known as wavelength dispersion. The first optical-fibre communications installations used LEDs (Chapter 10). These had a spectral width of about 35 nmcentred upon awavelength of 0.82 mm, which sets a severelimit on the rateofdata transmission and forced the introduction of lasers with a spectral width of 2 nm or less for signal sources. Unfortunately, even with completely monochromatic light, pulse spreading can still occur, due to the fact that the radiation can take various modes (paths) through the fibre. A ray that travels along the axis of a fibre will travel less far than one which is reflected many times on its journey (Figure 2.25). (In fact, the dispersion that results cannot be properly understood in terms of the transmission of light rays, and the various modes are better described in terms of the allowed wave patterns that can travel down the fibre.) The resultant pulse broadening, due to the various modes present, is called modal (or intermodal ) dispersion. Inorder toovercome modal dispersion a number of different fibre typeshave evolved. The earliest fibres were called stepped-index multimode fibres. These fibres have a large core region, allowing many modes to propagate (Figure 2.26a). The ray labelled H in Figure 2.26a is known as a high-order mode and the ray L is a low-order mode. Stepped-index multimode fibres are easy to make and join, but have a lower performance than those described below. Stepped-index fibres are adequate for short-distance communications but not for medium- or long-distance links. The first advance on stepped-index fibres was the GRIN fibre. In this design, the refractive index of the fibre varies smoothly from high at the centre to low at the periphery of the core region. The refractive index gradient means that light travels faster and faster as it approaches the edge regions of the fibre. The velocity of mode A will be fairly constant, while thevelocity of mode B will vary smoothly from lowest at the fibre centre to greatest near to the fibre edge (Figure 2.26b). The differences in path length between high-order and low-order modes h us aetaeld1km 1 travelled have pulse the 83 a ah hyaeral lentv ih aepatterns wave light alternative really are they paths ray 2.25 Figure (t by given is pulse light a of dispersion The (d) c). (b, fibre optical an 2.24 Figure oor u oRfato n Dispersion and Refraction to Due Colours h rda iprin(rsraig fasre fiiilysaplgtple a,a hymv along move they as (a), pulses light sharp initially of series a of spreading) (or dispersion gradual The h loe ah htlgtcntk hog notclfir r aldmds lhuhdanas drawn although modes; called are fibre optical an through take can light that paths allowed The cladding core (d) (c) (b) (a)

Amplitude 0.5 1.0 00 t 0 0.5 1.0 t x ray 2(mode2) x t 0 Distance ;te5 mltd ekwdh after widths peak amplitude % 50 the ); ray 1(mode1) Colour and the Optical Properties of Materials 84

~100 μm cladding H core L

(a) ~140 μm

~65 μm cladding A core B

~125 μm (b)

~10 μm

cladding core

Refractive index n Figure 2.26 Types of optical fibre: (a) stepped-index fibre; (b) GRIN fibre; (c) monomode fibre are thus minimized by this velocity variation. GRIN fibres reduce modal dispersion by a factor of about 25 to about 1 ns km 1. Even this improvement is insufficient for long-distance communications. For best results monomode fibres are required (Figure 2.26c). The number of possible modes is reduced simply by reducing the diameter of the core. When the core diameter reaches 10 mm or less, only one mode can propagate and, in principle, modal dispersion is zero for these fibres. Monomode fibres, therefore, have a high performance but are harder to make and join. Although fibres incommercial use are made ofsilica glass,itis not perfect.The dispersion is lowest at 1.3 mm, buttheminimumattenuationoccursat1.5 mm,leadingtosomesacrificeofperformanceirrespectiveofthesignal wavelengthchosen.Thesearchfornewmaterialstoresolvethisconflictcontinuesinmanyresearchlaboratories.

2.10 Negative Refractive Index Materials

2.10.1 Metamaterials

For all practical purposes involving light, the refractiveindex of a transparent solid is positive.However, there is no theoretical reason why this should always be so, and much current interest centres upon materials which display a negative refractive index. These materials are often called negative-index materials (NIMs). In fact, it 85 Colours Due to Refraction and Dispersion has been known for any years that the refractive index of many materials is negative for X-rays. However, the effect with X-rays is minute and the current interest is in materials that show an appreciable negative refractive index at frequencies near to those of the visible spectrum. A good deal of research is at present underway in order to create structures that will lead to a negative refractive index. The current awareness of the potential of negative refractive index materials started with an exploration of their use in the construction of perfect lenses (see Pendry in this chapter’s Further Reading). The problem in creating a NIM is implicit in Equation 2.8, Section 2.3:

2 n ¼ er ð2:8Þ

Now this is an approximation valid for transparent phases at optical frequencies. The more correct equation is:

2 ¼ e m n r r where er is the relative permittivity and mr the relative permeability of the material. Roughly speaking, this can be interpreted by saying that both the electric and magnetic dipoles in the material contribute to the refractive index. For all ordinary transparent materials er is positive and mr is approximately unity, leading to Equation 2.8. This is because, at optical frequencies, the light electrons are able to keep up with the electric field component (Section 2.3), while the magnetic dipole contribution, which arises with unpaired electron spins and electron orbital moments, is unable to react to the rapidly changing magnetic field component. This results in a value of approximately unity for mr and leads to the contention in Chapter 1 that only the electric field component of the electromagnetic wave needs to be considered when optical properties are paramount. To make a negative index phase for any particular frequency range, it is ideally necessary to obtain a material in which both er and mr are negative. It turns out that it is easy to obtain a material with a negative value of er. Metals containing free electrons, especially the noble metals, copper, silver and gold, show this, as do some ferroelectric compounds over some ranges of the electromagnetic spectrum. The problem is to obtain a matching negative value of mr. This is not possible in a single natural material, but has been possible in artificially created composite structures. These combinations are known as metamaterials. A metamaterial is a periodic structure (like a crystal) with artificially designed component ‘atoms’ that give the material the desired properties. The important structural features must be smaller than the wavelength of the electromagnetic radiation that the material is designed to influence, and so it is not surprising that the first negative index metamaterials were designed to act on microwaves with wavelengths of the order of centimetres. An early metamaterial ‘crystal’ was composed of copper wires in a cubic grid, together with open copper rings (split-ring resonators) at the nodes (Figure 2.27). In this design, the copper wires provide the negative permittivity component and the split rings the negative permeability component. Thewavelength of radiation that will respond to this negative index structure is approximately the same as the spacing of the pairs of split-ring resonators. Many other designs of metamaterial are currently being explored, including ‘fishnet’ structures. In this design, a periodic array in which thin films of silver sandwich a thin layer of an insulator such as alumina (Al2O3) provides a negative permeability (Figure 2.28a). When two sets of these arrays are arranged perpendicular to each other a fishnet structure is formed (Figure 2.28b). One part of the net provides the negative permeability ‘atoms’ and the continuous threads of the net provide the negative permittivity wires. These structures can be fabricated with spacings such that they respond to optical-frequency radiation. Photonic crystals (Chapter 6), structures made up of ordered ‘crystal-like’ arrays of pores or similar ‘defects’, can also be designed to show a negative refractive index at optical frequencies. Colour and the Optical Properties of Materials 86

copper wire split ring resonator

Figure 2.27 Schematic diagram of a metamaterial made up of pairs of metallic split rings mounted on copper wires to form a crystal-like structure that shows negative refractive index for radiation with a wavelength approximately equal to the spacing of the rings

(a)

silver insulator silver

substrate

(b)

Figure 2.28 (a) An array of silver strips separated by an insulating layer is able to form a negative refractive index solid. (b) Two arrays as in (a) make up a ‘ﬁshnet’ metamaterial 87 Colours Due to Refraction and Dispersion

2.10.2 Superlenses

Negative refractive index materials are being explored in a number of contexts, including ‘invisibility’ or cloaking screens, which act so as to divertincident radiation in such away as to render the object ‘invisible’ to an observer. However, the application of most relevance to the subject matter of this book concerns superlenses or hyperlenses. Normal lenses, those in telescopes, microscopes, cameras and other optical instruments, have a resolution limited to approximately thewavelength of the imaging light, due to diffraction (Section 6.4). That is to say, anyfeature smaller than thewavelength of light cannot be imaged optically. (Image formation can also be treated in terms of information theory, in which case the information content of the image is limited to detail of the order of the wavelength of light or greater.) This has always been an impediment for scientists who wish to understand physical, chemical or biological processes at the molecular or cellular level and led directly to the invention and use of, among other techniques, electron microscopy, because the wavelength of an electron beam is considerably less than that of a light beam. The limitation on imaging detail is also of considerable importance with respect to data storage and transmission, as the desire to make ever smaller components on silicon chips and related circuitry has continually been hampered by this imaging constraint. The use of negative refractive index materials has allowed these barriers to be overcome, at least in part, and lenses able to image details considerably smaller than the wavelength of the incident light have been produced. Lenses that can bypass the diffraction limit are called ‘superlenses’ or ‘hyperlenses’. The ﬁrst remarkable result with respect to this phenomenon is that a plane slab of negative refractive index material can actually form an image. This is because Snel’s law continues to operate, but the refracted beam now lies on the same side of the normal as the incident beam rather than the opposite side, which can be seen by applying Equation 2.2:

n1sin1 ¼ n2sin2

When n2 is negative, 2 will be negative (Figure 2.29a and b). This in turn implies that a slab of negative refractive index material can form an image lens shapes are not needed (Figure 2.29c). The second remarkable point is that use of a negative refractive index slab can form an image with a superior resolution to that of a conventional lens. A conventional lens forms an image using progressive waves, the normal light waves. The detail that these images contain is limited by diffraction and is roughly of the order of the wavelength of the light used (Chapter 6). Now, evanescent waves carry more detailed information and if they can also contribute to image formation then the resultant ‘information content’ or resolution should be better than that of progressivewaves alone, and be able to reveal detail that is smaller than the wavelength of the light. This is possible and comes about in the following way. A slab of NIM acts so as to increase the amplitude of an evanescent wave in an exponential fashion, which is opposite behaviour to that of a normal material (Figure 2.29d). This means that the image formed by the slab, as described above, can also include information provided by the evanescent wave and thus may have a superior resolution to that shown by an ordinary lens. The question, therefore, is can a slab of NIM be used in this conceptually simple way? The answer is yes, and this (third) remarkable feature of NIMs is that, despite the limitations on the fabrication of these substances, a simple layer of silver metal will act as a superlens. This comes about in the following way. It is found that the equations for the reﬂection and transmission of light are independent of the permeability of the solid for light that is polarised in the plane of incidence (the p-wave or transverse magnetic wave; Chapter 4). As pointed out above, silver is a metal that shows a negative refractive index in the optical region that arises from the permittivity component of the refractive index. Hence, use of the correctly polarised incident light should give the required sub-wavelength resolution, a feat that has been successfully achieved. Colour and the Optical Properties of Materials 88

(a) (b)

θ θ 1 1

air air

n positive n negative θ 2 θ 2

ray path if n2 is positive

evanescent wave if n2 is +ve Figure 2.29 (a) Snel’s law for a normal material. (b) Snel’s law for a NIM. (c) Image formation by a slab of NIM. (c) Image formation by a slab of NIM. (d) The amplitude of an evanescent wave in a NIM

(a) microscope

Silica

magnified image Hyperlens: object 16 (Ag / Al2O3) layers

Cr layer

illumination (365 nm)

Figure 2.30 (a) Magnifying hyperlens composed of 16 Ag/Al2O3 cylindrical layers; schematic. The object is inscribed upon a Cr layer and the magniﬁed image can be viewed and photographed using a microscope. (b) A sub- diffraction limit image obtained with the lens [(a) and (b) from Science, Far-Field Optical Hyperlens Magnifying Sub-Diffraction-Limited Objects by Z. Liu, et al., 315, 5819, 1686 Copyright (2007). Reprinted with permission from AAAS.] 89 Colours Due to Refraction and Dispersion

Figure 2.30 (Continued)

There is a drawback to this technique. The image is formed close to the foil, in a region where the evanescent wave is appreciable. This limits usefulness considerably, but can be of value in the fabrication of nano-scale devices using contact methods of detail transfer so that the distance between the object and image is minimal. This limitation can be bypassed using more complex metamaterial design. A curved metamaterial, composed of alternating cylindrical layers of alumina and silver, acts as a magnifying lens (Figure 2.30a). The image is still formed close to the exit surface of the lens, but the image is magnified by the material curvature. Provided that the magnification is greater than the diffraction limit of an ordinary optical microscope, it can then be magnified optically at will. This technique has allowed images of features about 1/10 the wavelength of the imaging illumination to be photographed directly (Figure 2.30b). This rapidly advancing field is at a very exciting stage.

Further Reading

The effect of refraction on vision in water, especially from the point of view of a fish or of a fisherman, is described by J. D. Walker, Sci. Am. 250 (March), 108 113 (1984). Details of archer fish vision are given by S. Temple, N. S. Hart, N. J. Marshall, S. Collin, Proc. R. Soc. Lond. Ser. B 277, 2607 2615 (2010). Colour and the Optical Properties of Materials 90

Evanescent waves are introduced clearly in N. Braithwaite (ed.), Electromagnetism, Book 3, Electromagnetic Waves, The Open University, Milton Keynes, 2006. See also W. Knoll, Mater. Res. Soc. Bull. 16 (July), 29 39 (1991). The relationship between refractive index, polarisability and the Gladstone Dale (and other) equations is given in F. D. Bloss, Crystallography and Crystal Chemistry, Holt Rinehart and Winston, New York, (1971), Chapter 11. R. E. Newnham, Structure Property Relations, Springer, Berlin, 1975, Chapter 5. J. A. Mandarino, Can. Mineral. 14, 498 502 (1976); 16, 19 174 (1978); 17, 71 76 (1979); 19, 441 450 (1981). Animal eyes and the relevance of graded-index optics are described in L. P. Lee, R. Szema, Science 310, 1148 1150 (2005). The moth-eye antireflection surface structure is described by P. Vukusic, J. R. Sambles, Nature 424, 852 855 (2003). The physics of the rainbow is given by C. B. Boyer, The Rainbow: From Myth to Mathematics, MacMillan Education Ltd, Basingstoke. (1987). (This gives a very full history of the rainbow and its explanations, from ancient times. The first edition (1959) only has black-and-white photographs. The 1987 edition has colour illustrations.) V. Khare, H. M. Nussenzveig, Phys. Rev. Lett. 33, 976 980 (1974). H. M. Nussenzveig, Sci. Am. 236 (April), 116 127 (1977). J. D. Walker, Am. J. Phys. 44, 421 433 (1976); Sci. Am. 237 (July), 138 144 (1977). A discussion of rainbows, halos and other dispersion colours found in nature is given by D.K.Lynch,W.Livingston,ColorandLightinNature,CambridgeUniversityPress,Cambridge,1995,Chapter4. Fibre optics is covered in detail by J. Hecht, Understanding Fibre Optics, 3rd edition, Prentice Hall, Upper Saddle River, NJ, 1999. The evolution of fibre-optic communications can be appreciated by reading the following series of articles: W. S. Boyle, Sci. Am. 237 (August), 40 48 (1977). A. Yariv, Sci. Am. 240 (January), 54 62 (1979). M. G. Drexhage, C. T. Moynihan, Sci. Am. 259 (November), 76 81 (1988). E. Desurvire, Sci. Am. 266 (January), 96 103 (1992). G. Stix, Sci. Am. 284 (January), 68 73 (2001). D. J. Bishop, C. R. Giles, S. R. Das, Sci. Am. 284 (January), 74 79 (2001). D. J. Blumenthal, Sci. Am. 284 (January), 80 83 (2001). Negative refractive index materials, superlenses and hyperlenses can be reviewed by consulting J. B. Pendry, Phys. Rev. Lett. 85, 3966 3969 (2000). J. B. Pendry, D. R. Smith, Sci. Am. 295 (July), 43 49 (2006). Various authors in Mater. Res. Soc. Bull. 33 (October), (2008). Z. Liu, H. Lee, Y. Xiong, C. Sun, X. Zhang, Science 315, 1686 (2007). There are a number of demonstrations of relevance to this chapter, including refraction by prisms, water droplets and NIMs, available at http://|demonstrations.wolfram.com/index.html. 3 The Production of Colour by Reflection

. Why are soap bubbles coloured? . How do antireﬂection coatings on lenses function? . How can perfect mirrors be made from transparent materials?

Reflection is a commonplace phenomenon and the appearance of a solid is often dominated by reflection. As well as modifying the perceived colour of a body in terms of surface gloss, reflection as such can give rise to a surprising range of colours. The most vivid of these are associated with the presence of reflection by thin transparent films. Bright colours are often seen in soap bubbles, and close examination of transparent insect wings shows that these can show areas which are beautifully coloured. Casual observation also reveals that the colours have a metallic aspect (due to a considerable specular component) and seem tovary with the direction of viewing and with the thickness of the film. They are said to be iridescent. In this chapter the origin of these and other colours due to reflection is explored. Attention is confined to reflection by more or less transparent insulating solids (dielectrics in older literature). Metals are considered in a later chapter. However, recall that the refractive index varies with wavelength, and metals are transparent at some wavelengths and in these circumstances the conclusions of this chapter will then apply. It is necessary to mention that polarisation of light is important in reflection. In this chapter we are concerned mainly with the colours produced by unpolarised sunlight, and the refinements needed to account for the polarisation of thewaves are considered in Chapter 4. This objective is aided by mainly considering light falling perpendicularly onto surfaces, for which the polarisation direction of the wave becomes redundant.

Colour and the Optical Properties of Materials Richard J. D. Tilley 2011 John Wiley & Sons, Ltd Colour and the Optical Properties of Materials 92

3.1 Reﬂection from a Single Surface

3.1.1 Reﬂection from a transparent plate

When light falls onto a smooth, thick transparent plate such as a slab of glass, so that the lower surface can be ignored, some ofit will be reflected and some transmitted (Figure 3.1). For a smooth metal plate almost all of the light will be reflected, as the amount of transmitted light is negligible. For both materials, thewell-known law of reflection is:

1 ¼ 3 where 1 (also i or i) is the angle of incidence and 3 (also r or r) the angle of reflection. The plane of incidence contains the incident ray, the reflected ray and the normal to the reflecting surface. In Figure 3.1 this is the plane of the page. The amount of light reflected from such a surface depends upon the polarisation of the light (Chapter 4). For a polished, thick plate and light at normal incidence (i.e. perpendicular to the surface) the polarisation can be ignored and the coefficient of reflection r, defined such that if a wave of amplitude E0 falls upon the surface then the amplitude of the reflected wave is rE0, is given by: n n r ¼ 0 s n0 þ ns where n0 and ns are the refractive indices of the media on the two sides of the boundary in the direction in which the light travels (Figure 3.2). The eye detects irradiance changes rather than amplitude changes, and so it is the more convenient to work with the reflectivity or reflectance R: n n 2 R ¼ r2 ¼ 0 s n0 þ ns

E2 This is because the irradiance I0 is proportional to the square of the amplitude 0. The reﬂected irradiance R(I0) 2E2 (Figure 3.2) is then proportional to r 0.

θ1 θ3

air

θ2 glass

Figure 3.1 Light falling on a transparent plate such as a slab of glass will be partly reflected and partly refracted. u u The angle of incidence 1 will be equal to the angle of reflection 3 93 The Production of Colour by Reflection

ε ε 0 r 0 I0 R I0

Figure 3.2 Reflection of a beam of light perpendicular to a transparent surface. The amplitude of the incident E E beam is 0 and incident irradiance is I0. The reflected amplitude will be given by r 0 and the reflected intensity by RI0. The angles have been exaggerated for clarity

Remember that because n depends upon wavelength, the coefficient of reflection and the reflectivity will vary across the spectrum. When light travelling through a medium of low refractive index (such as air) is reflected at the surface of a substance of higher refractive index (such as glass), r is negative. This signifies a phase change of p radians on reflection, which means, in terms of a light wave, that a peak turns into a trough, and vice versa, whenever the refractive indices at the interface are in the sequence low/high (Figure 3.3). The reflectivity R for a transparent plate of refractive index ns in air is: 1n 2 n 1 2 R ¼ s ¼ s ð3:1Þ 1 þ ns ns þ 1

n n 0 1 incident ray

phase change ray after reflection

Figure 3.3 A phase change of p radians is introduced in a ray reﬂected at a surface of higher refractive index, which means that a peak changes to a trough and vice versa Colour and the Optical Properties of Materials 94

For ordinary glass, with a refractive index of approximately 1.5, R is about 0.03 0.04; that is, about 3 4 %. Although this may seem to be a rather insignificant amount, it is noticeable in everyday life. Reflections from windows and from the glass over a painting are frequently annoying. Moreover, it is too high for specialist purposes, such as high-performance lenses and optical components, so these are given antireflection coatings, discussed below. In addition, this small degree of reflectivity turns out to be an essential component in the production of colour through interference by thin films.

3.1.2 Data storage using reﬂection

In essence, the active data storage layer (or layers) on CDs, DVDs, HD-DVDs and Blu-Ray discs is reflective. Data is stored by making small dots on the recording layer that have a different reflectivity to the background. The writing process involves decreasing the reflectivity and the reading process involves detecting these reflectivity differences. The discs that contain permanent read-only data, such as those sold in stores, are produced with physical pits in the surface. The pit will have a different reflectivity to the higher surrounding plateau. The smaller the pit, the more data can be stored on the disc. Pit size is more or less controlled by the wavelength of the light used to record the data. The CD, introduced in 1983, used infrared wavelengths (780 nm) generated by semiconductor lasers (Section 10.9). The pit size minimum was 0.83 mm and the track pitch (the separation between lines of pits) was 1.6 mm. Great effort was expended in moving towards red light, wavelength 650 nm, which allowed for smaller pit dimensions. Thus, the DVD, introduced in 1996, using this wavelength, has a minimum pit size of 0.4 mm and a track pitch of 0.74 mm. Recently, there has been a move to blue light, wavelength 405 nm, allowing for smaller pits and greater data storage, resulting in the transition to HD-DVD and Blu-ray technology. Recordable CDs (CD-R) use similar reflectivity differentials to record data. In this instance, a writing laser in the computer marks small spots in a layer of polymer containing a dye. These spots are detected as changes in reflectivity when scanned by the reading laser. Rewritable CDs (CD-RW) make marks in a layer of a crystalline Ag In Sb Te alloy. The exact composition of the alloy is carefully chosen so as to yield optimum recording and erasing facilities under the influence of the lasers normally present in home computers. The alloy is crystalline as prepared. When heated by a pulse from the writing laser beam, to a temperature of approximately 700 C, the alloy melts locally and cools too rapidly to crystallise, thus ending in an amorphous (glass-like) state. This has a lower reflectivity than the surrounding crystalline surface. Erasure is carried out by heating the amorphous data spot with a laser pulse to about 200 C, which allows solid-state crystallisation to occur, restoring the original surface reflectivity. The writing process can then be carried out once more. The marks on a CD or DVD are comparable in size to the wavelength of light. This means that light rays can interfere and produce iridescent colours. These are discussed in terms of diffraction in Chapter 6.

3.2 Interference at a Single Thin Film in Air

Part of a monochromatic beam of light incident upon the top surface of a homogeneous thin film of refractive index n will be reflected. The remainder will enter the film and be repeatedly reflected from the bottom surface and the underside of the top surface. At each reflection some of the light will escape to produce additional reflected and transmitted rays (Figure 3.4a). The consequence of this repeated reflection and transmission is to produce the bright colours seen on thin films of many types. 95 The Production of Colour by Reflection

(a) 1 2 3

air

(b) 2 1 3

air

d n

top surface (c)

ray 1 (incident ray )

ray 2 (reflected)

ray 3 (refracted and reflected)

out of step = retardation

Figure 3.4 The reflection and transmission of a ray of light incident on a transparent film in air. (a) A number of reflected and transmitted beams occur due to repeated reflection at the top and bottom faces of the film. (b) At normal incidence (the angles of incidence and reflection have been changed from 90 for clarity), ray 3 will have travelled further than ray 2 by a path difference of 2nd. (c) The waves making up rays 2 and 3 will be out of step due to the combined effects of a phase change on reflection and the path difference Colour and the Optical Properties of Materials 96

3.2.1 Reﬂection perpendicular to the ﬁlm

When the light beam is perpendicular to the surface the complexity of dealing with polarisation is avoided, so that the analysis of the phenomenon is simplified. In this case, part of the light seen by an observer above the film will have been reflected at the top surface (ray 2). In addition, somewill have travelled through the film and been reflected from the bottom surface before reaching the observer (ray 3) (Figure 3.4b). As the reflectivity is rather small, about 4 % for a glass surface, the first reflected ray and the first ray transmitted through the glass and then reflected from the lower surface are of most importance. For the present, the other transmitted and reflected rays will be ignored. Because of the difference in the paths taken by the two rays, thewaves will be out of step. In addition, because ray 2 is travelling through a medium of low refractive index and is reflected at a low high refractive index interface a wave peak will turn into a trough and viceversa. This will not happen to ray 3 because it is reflected at a high low refractive index surface (Figure 3.4c). On leaving the thin film the waves making up rays 2 and 3 can now interfere, which will cause the film to look either dark or bright. The effect is easily understood. Ray 3 will have travelled further than ray 2 by a twice the film thickness, 2d. However, the physical thickness does not give the mismatch between the wave crests and peaks of these two rays. This is given by the additional optical path travelled by ray 3, in this case 2nd, where n is the refractive indexof the film and d is the physical thickness. The path difference (or retardation) between rays 2 and 3 is equal to the optical path difference between the two rays, so that:

p ¼ 2nd

The appearance of a thin film when viewed by reflection at normal incidence will depend upon this path difference. If the path difference p considered in isolation is equal to an integral number of wavelengths then the waves will be exactly in step as they travel away from the surface. Adding in the phase change of half a wavelength for ray 2 will make it out of step with ray 3 by this amount as they leave the surface. The consequence is that destructive interference will occur between ray 2 and ray 3. The film will, therefore, appear dark. In general, the film will appear dark under irradiation with light of wavelength l0 in air when:

p ¼ 2nd ¼ ml0 ðm ¼ 1; 2; 3; ...Þ minimum ðdarkÞ

In a similar way, a path difference p between ray 2 and ray 3 equal to a half-integral number of wavelengths will cause the two rays to be exactly out of step. Adding in the half-wavelength phase change for ray 2 will make them exactly in step. The ﬁlm will then appear bright, because constructive interference will occur. In general, the ﬁlm will appear bright when:

p ¼ 2nd ¼ðm þ 1Þl ðm ¼ 1; 2; 3; ...Þ maximum ðbrightÞ 2 0

At other path differences the film will appear to have an intermediate tone, depending upon the exact phase difference between ray 2 and ray 3. When a tapered or wedge-shaped film is viewed by monochromatic light at normal incidence, some thicknesses will be appropriate for constructive interference and some for destructive interference. The film will then appear to be crossed by a series of bright and dark bands (Figure 3.5a and b). When the thickness of the 1 l ¼ film is considerably below 2 0 the film will appear dark, because the path difference in the film, p 2nd, will not be sufficient to counteract the change of phase of the ray reflected from the upper surface, and destructive interference occurs. As the film thickness increases, it will eventually reach the stage where the destructive 97 The Production of Colour by Reflection

(a) (b) 0 dark

λ/2 bright

1λ dark

3λ/2 bright

2λ dark

5λ/2 bright

3λ dark

7λ/2 bright

4λ dark

Figure 3.5 Interference in a wedge shaped film: (a) film profile; (b) bright and dark reflection bands resulting from the interference of monochromatic light viewed normal to the wedge from above; (c) an air gap between two transparent plates behaves in a similar way to a wedge interference is replaced by constructive interference, and a bright band appears, centred at p ¼ 1l . Thereafter, 2 0 bright and dark bands will succeed each other, each at an interval of p ¼ 1l . 2 0 Precisely the same effect will be obtained for an air wedge between two inclined transparent plates (Figure 3.5c). Bright fringes will be observed as at intervals of:

p ¼ 2nd ¼ðm þ 1Þl ðm ¼ 1; 2; 3; ...Þ maximum ðbrightÞ 2 0

In cases where the wedge or air gap is not uniform, the fringes follow contours of equal thickness. This gives a dynamic impression of surface contours.

3.2.2 Variation with viewing angle

When the light beam is at an angle to the surface, polarisation effects become important. These are not severefor angles of incidence of up to approximately 25 to the surface (see Figure 4.4) and will be ignored here. The path difference p between rays 2 and 3 becomes (Figure 3.6):

p ¼ 2nd cos2

The analysis now follows that given in the previous section. If p is equal to a whole number of wavelengths then the ﬁlm will appear dark, due to the combined effect of path difference and change of phase of ray 2 on reﬂection at the surface. Thus: Colour and the Optical Properties of Materials 98

1 2 3

θ air 1

θ 2 n θ 2 d

air

4 Figure 3.6 The reﬂection of a ray of light incident on a transparent ﬁlm in air. The path difference between rays 2 u and 3 is 2nd cos 2

p ¼ 2nd cos2 ¼ ml0 minimum ðdarkÞ

For the same reason, if the path difference turns out to be a half wavelength, reinforcement will occur and we ﬁnd:

p ¼ 2nd cos ¼ðm þ 1Þl maximum ðbrightÞ 2 2 0

When viewed in monochromatic light, the regions of the ﬁlm which look bright or dark will change with the viewing angle. If the light is normal to the ﬁlm, then cos2 ¼ cosð0Þ¼1 and the equations reduce to thosegiven in the previous section. Naturally, the same is true of wedges and wedge-shaped air gaps.

3.2.3 Transmitted beams

The analysis given in the preceding section for interference between the reﬂected beams can be repeated for the transmitted beams (Figure 3.7). The path difference between beams 4 and 5 is:

p ¼ 2nd cos2

In this case, there is no extra phase change, as all reﬂections take place at the high low refractive index interfaces. Thus, when:

p ¼ 2nd cos2 ¼ ml0 maximum ðbrightÞ there will be constructive interference and a maximum in transmitted intensity. When:

p ¼ 2nd cos ¼ðm þ 1Þl minimum ðdarkÞ 2 2 0 99 The Production of Colour by Reﬂection

1 2 3

air

θ 2 d n

4 5 Figure 3.7 The transmission of a ray of light incident on a transparent ﬁlm in air. The path difference between u rays 4 and 5 is 2nd cos 2

there will be destructive interference and a minimum in the transmitted intensity. This is converse to the case of reflection, so that a dark reflection band corresponds to a bright transmission band. The two patterns are complementary. Note, however, the appearances of the patterns are somewhat different. In the case of reflection, the intensities of rays 2 and 3 are similar. In the case of transmission, ray 4 will have approximately 96 % of the incident intensity, while ray 5, which has suffered two reflections, will have about 1/1000 of this value.

3.3 The Colour of a Single Thin Film in Air

Although a discussion of monochromatic light is helpful so as to understand the physical processes taking place on reflection at a thin film, we are really much more interested in what the appearance of the film will be in daylight. When the film is viewed in white light, the same reflection and interference discussed above will occur, except that the effects of all of the different wavelengths present must be added. These interference effects lead to intense colours, familiar in soap films, oil films on water puddles and thin flakes of minerals which can glint with bright colours in sunlight (Figure 3.8). For example, violet light with a wavelength of 400 nm will reflect a maximum of intensity when the film ¼ ð þ 1Þl produces a path length difference or retardation p ( 2nd)of m 2 , i.e. 200 nm, 600 nm, 1000 nm and so on (Figure 3.9). Of course, these values of the path difference will not give a maximum for the other wavelengths present in white light. In fact, for a wavelength of 600 nm, there will be a minimum of intensity for the same retardation of 600 nm (Figure 3.9). In order to determine the reflected colour of a thin film whenviewed in white light it is necessary to add up all of these contributions over all of the values of the wavelength present. It is seen that there is a large contribution from the 400 nm wave. The contribution from succeeding waves decreases until at a wavelength of 600 nm there is no contribution at all. Thereafter, a small contribution is obtained from wavelengths of 650 and 700 nm. (In reality, a continuum of wavelengths occurs between 400 and 700 nm of course. Here, just sevenwavelengths are used as an illustration.) The overall colour perceived will be the sum of all of these. Because of the dominance of the 400 nm contribution the film will appear to be a violet blue colour. Thereafter, the perceived colour will vary as the film thickness increases or decreases, as certain colours are either reinforced or cancelled. The sequence of colours seen will repeat in a cyclical fashion as the film thickness changes (Figure 3.10). Each sequence of spectral colours is called an order, which starts with the first order for the Colour and the Optical Properties of Materials 100

Figure 3.8 Interference colours in thin films: (a) soap bubbles; (b) flakes of molybdite (molybdenum trioxide, MoO3). Both films are viewed in reflected white light thinnest of films. A new order begins every 550 nm of retardation (¼2nd). The colour of a thin film viewed by reflection in white light is given in Appendix A3.1. Ultimately, strong interference effects will be lost. This is because ordinary white light is emitted in bursts which undergo a sudden change of phase every 10 8 s or so. When the film is very thin the two rays which interfere come from within the same burst and interference effects are noticeable. With thicker films, interference takes place between different bursts of light and the interference effects are weaker. This results in films showing mainly pale pinks and greens in the fourth and fifth orders. With even thicker films all interference effects are smoothed out and colours are no longer apparent to the eye. Since the fraction of incident white light which is reflected is coloured, it follows that the transmitted light will be depleted in this colour and the colour seen will, therefore, be the additive complementary colour to that strongly reflected. These are listed in Appendix A3.1. If the angle of viewing is not perpendicular to the film then the retardation changes slightly. The correct expression for this is:

p ¼ 2nd cos2 as described before. This formula indicates that, as the viewing angle moves away from perpendicular to the film, the colour observed will move towards lower retardation. Thus, for example, second-order orange red will change towards green and blue (Figure 3.10). (But note that, at all angles except for perpendicular viewing, polarisation will also occur and be important.) This discussion explains the familiar colours of soap films seen in air. These are best seen if the film is viewed against a black background, which prevents the effects being masked by other reflections. As the thickness of the films varies, due to water flow within the films themselves, the colours change in a dramatic and beautiful way. A draining film has a number of possible equilibrium thicknesses. The thinnest, with a thickness of about 6 nm, gives a black film called Newton’s black film. The ways in which a thinning film produces rivers and streams of black in a coloured surrounding film are legion, and no two casually produced films seem to drain in 101 The Production of Colour by Reflection

g 700 nm

f 650 nm

e 600 nm

d 550 nm Intensity

c 500 nm

b 450 nm

a λ = 400 nm

400 800 1200 1400 Retardation / nm Figure 3.9 The intensity (in arbitrary units) of light reﬂected from a single thin ﬁlm in air at various wavelengths plotted as a function of the retardation – the optical path difference between the two interfering beams

the same way. If the ﬁlms are formed on a wire frame, then the transmitted and reﬂected colours can be compared.

3.4 The Reﬂectivity of a Single Thin Film in Air

Interference and colour, as just discussed, should be differentiated from reflectivity. It could be that a certain colour, say red, is produced by interference effects in a film, but whether the colour is readily seen will depend upon the reflectivity of the film for this wavelength. The reflectivity of a thin film in air will be different from that for a thick plate (Equation 3.1), as interference effects from the bottom surface also need to be considered. However, the polarisation of the light will be important and can only be neglected when the light is incident perpendicularly to the surface of the film. Colour and the Optical Properties of Materials 102 white yellow orange-red yellow green violet-blue blue violet-blue grey orange Intensity

1st order 2nd order

100 500 1000 Retardation / nm Figure 3.10 The total intensity (in arbitrary units) reflected from a thin film in air illuminated by white light as a function of the retardation (the optical path difference) of the film. The approximate colours observed by eye are indicated

For light of a single wavelength at normal incidence the reﬂectivity of a homogeneous transparent thin ﬁlm is given by: 2r22r2cos2d R ¼ 1 1 2 d þ 4 1 2r1 cos2 r1 where

n0nf r1 ¼ n0 þ nf n0 is the refractive index of the surrounding medium (often air, with n0 ¼ 1.0) and nf is the refractive index of the ﬁlm:

p½ p d ¼ 2 d ¼ 2 nf d l l where [d] is the optical thickness of the film and d is the physical thickness. The reflectivity is found to vary in a cyclic fashion from zero for values of [d] equal to 0, l/2, l, etc. to a maximum (of approximately 0.24 for nf ¼ 1.7) for values of [d] given by l/4, 3l/4 and so on. Because the refractive index of the film is a function of wavelength, the reflectivity will also vary across the spectrum.

3.5 The Colour of a Single Thin Film on a Substrate

The behaviour of a single thin film on a substrate is similar to that discussed for the case of a single thin film in air. Thus, a thin transparent film on a substrate would be coloured when viewed in white light. To analyse this situation, it is necessary to take into account any change of phase that might occur on reflection at the back surface of the film. The actual hue perceived will be found by a summation of all of the reflected intensities, as was discussed earlier. 103 The Production of Colour by Reflection

If the substrate has a lower refractive index than the film on the surface then the treatment will be identical to that for a thin film in air, from the point of view of interference effects. The reflected colours observed when the film is viewed at normal incidence in white light will be the same as those listed as ‘colour reflected’ in Appendix A3.1. (The transmitted colours are normally absorbed by the substrate.) If the refractive index of the substrate is greater than that of the film then a phase change will be introduced at both the air film interface and the film substrate interface. In this case the reflected colour seen at normal incidence when viewed in white light will be the complementary colour to that just described, listed as ‘colour transmitted’ in Appendix A3.1. For example, if it is necessary to estimate the thickness of a film of SiO2 grown on the surface of a single crystal of carborundum (SiC, silicon carbide) use the ‘colour transmitted’ list, because the refractive index of silicon carbide is greater than that of silica. In fact, silicon carbide is strongly absorbing over the visible spectrum and when first grown the crystals of carborundum are a shiny black. However, they soon take on a wide variety of attractive iridescent colours because of surface oxidation, which produces thin silicon dioxide surface films in a wide variety of thicknesses (Figure 3.11). These films are protective and preserve the underlying material from further oxidation, so that the colours only change slowly over the course of time. Similar colours are seen on the surfaces of some metals due to oxidation. The film is usually of a transparent oxide, Al2O3 on Al, TiO2 on Ti, Ta2O5 on Ta and so on. When they form as a result of the metal being used as an electrode in an electrochemical cell the metal is said to be anodized. These films, if of the appropriate thickness, will be brightly coloured. (Note that some anodized films are made especially thick to protect the underlying

Figure 3.11 Colours due to white light interference in a thin transparent film of silicon dioxide (SiO2)on carborundum (SiC, silicon carbide). The colour variation is due to changes in film thickness Colour and the Optical Properties of Materials 104 metal. They are frequently coloured by the incorporation of dyes for decorative purposes. These colours are not thin film effects and arise from the dye molecules (Chapter 8).) Thin-film colours are also frequently seen when an oil layer covers a puddle of water on a road, where the refractive index of the oil is usually greater than that of the underlying water. These colours are enhanced to the eyeby the black road surface, which absorbs all light not reflected by the film. As the thickness of the oil changes in response to wind or water movement, the colours vary considerably. In fact, the eye is able to detect minute changes in surface appearance when a thin film is deposited upon a substrate. As surprising as this may seem, the fastest and simplest way to detect single atomic layers of graphene is to use reflected white-light optical microscopy. Graphene, which is being actively studied because of its remarkable electronic properties, is composed of a single layer of carbon atoms linked in a hexagonal array a single sheet from a graphite crystal, in fact. Graphene can be prepared by rubbing a graphite crystal over a smooth surface, a process called mechanical defoliation. A graphene layer on a such a substrate can be readily detected by eye because of the additional path difference introduced by the graphene layer, even though this is of only one atom in thickness, coupled with the fact that graphene absorbs a little of the incident light. Graphene sheets on silicon dioxide, for example, look transparent pale purple.

3.6 The Reﬂectivity of a Single Thin Film on a Substrate

The reflectivity of a single thin film deposited on a substrate, like that of a single thin film in air, depends upon the polarisation of the light, the film thickness and direction of the incident radiation. In the case of monochromatic illumination perpendicular to a homogeneous nonabsorbing thin film:

2r2 þ 2r r cos2d þ r2 R ¼ 1 1 2 2 þ d þ 2 2 1 2r1r2 cos2 r1r2 where

n0nf r1 ¼ n0 þ nf

nfns r2 ¼ nf þ ns n0 is the refractive index of the surrounding medium, nf is the refractive index of the ﬁlm and ns is the refractive index of the substrate. The expression for d is:

p½ p d ¼ 2 d ¼ 2 nf d l l where [d] is the optical thickness of the ﬁlm and d is the physical thickness of the ﬁlm. For values of [d] given by l/2, l,3l/2, etc. the equation reduces to:

ð Þ2 ¼ n0 ns R 2 ðn0 þ nsÞ 105 The Production of Colour by Reﬂection

This is identical to the equation for an uncoated surface. Thus, a layer of optical thickness l/2, etc. can be considered to be optically absent and the surface has normal uncoated reflectivity. This is an intriguing and useful result. It means that if a delicate surface is coated with a l/2 layer of a hard transparent material the surface will be protected without any effect on optical properties. For values of [d] given by l/4, 3l/4, etc. the reflectivity is given by: n2n n 2 R ¼ f 0 s ð3:2Þ 2 þ nf n0ns and the reflectancewill be eithera maximum or a minimum. This will depend upon whetherthe film has ahigher refractive index than the substrate or a lower refractive index than the substrate. When the refractive index of the film is between that of the surrounding medium and the substrate (n0 < nf < ns), the reflectivity will be a minimum. When the film has a higher refractive index than both the substrate and the surrounding medium (n0 < nf > ns), the reflectivity will be a maximum. As with a thin film in air, the value of the reflectivity will cycle with film thickness between a lower value at [d] equal to 0, l/2, l, etc. to a maximum for values of [d] equal to l/4, 3l/4 and so on. Because the refractive indices are a function of wavelength, the reflectivity will also vary across the spectrum.

3.7 Low-Reﬂection and High-Reﬂection Films

3.7.1 Antireﬂection coatings

We can easily use the above equations to see how thin films modify the reflectivity of a surface. Suppose that it is desired to make a nonreflective coating on a glass surface in air. (Such coatings are called antireflection (AR) coatings.) Equation 3.2 shows that if the value of nf lies between that of air and the glass then the reflectivity will be a minimum for a l/4 film. Putting R ¼ 0 in Equation 3.2 yields a value of the refractive index of the film which will give no reflection at all: p nf ¼ ns ð3:3Þ

For glass, ns is about 1.5, so the antireﬂecting ﬁlm must have a refractive index: p nf ¼ 1:5 ¼ 1:225

Very few solids have such a low index of refraction, and a compromise material often used is magnesium 1 fluoride, MgF2, for which n in the middle of the visible is 1.370 at 500 nm. This is not perfect, but does reduce the reflectivity from about 4 % down to about 1 % (Figure 3.12). The coating will actually be maximally antireflective for the design wavelength, which is the wavelength for which the AR coating is optimised and the amount of light reflected will increase for wavelengths on either side of the design wavelength and also for oblique angles of incidence. For camera lenses, which commonly use AR coatings, the design wavelength is usually near the middle of the visible spectrum, say 550 nm. Such films reflect violet and red more than yellow or green; an effect readily observed when a good coated camera lens is examined.

1 Note that MgF2 is not isotropic and the refractive index depends upon crystal direction (see Chapter 4). In this and similar cases, the coatings are made by evaporation and generally have a single effective refractive index. Colour and the Optical Properties of Materials 106

1 Reflectivity (%)

400 500 600 700 Wavelength / nm

Figure 3.12 The reflectivity of a quarter-wave film of a magnesium fluoride (MgF2) AR coating on a glass surface with a refractive index of 1.52. The design wavelength of the film is 550 nm and the beam is taken as perpendicular to the surface

3.7.2 Antireﬂection layers

Apart from their utility as surface coatings, AR layers are also important in a variety of applications. For example, the fabrication of an integrated circuit on a silicon chip involves one or more steps in which the material is exposed to light through a pattern called a mask. The mask is used to selectively illuminate areas on the chip which, after further processing, build into the array of transistors which manipulate data. The light actually interacts with a layer of substance called a photoresist. After illumination the photoresist employed is weakened in those areas which were exposed to light and these are subsequently dissolved away so as to reveal the underlying silicon, which can then be selectively doped or otherwise treated. The length of time of the exposure of the photoresist to light is critical to the success of the process. The desire to pack more and more transistors onto a chip has led to the drawing of ever finer detail onto the mask and the use of increasingly shorter wavelength light in the illumination steps. At present, the use of ultraviolet radiation is commonplace. The sharpness of the pattern produced on the silicon, and hence the number of transistors which can be placed onto the chip, is limited by diffraction (Chapter 6) and multiple reflections within the photoresist. The multiple reflections expose parts of the photoresist which should remain unexposed (Figure 3.13a). This has the effect of reducing the sharpness of the projected pattern and can also introduce spurious detail or defects. In order to combat this difficulty an AR coating can be applied between the silicon substrate and the photoresist (Figure 3.13b). The aim is to introduce a film of the correct thickness to ensure that successive rays reflected at the bottom surface of the photoresist are out of phase by l/2 so that destructiveinterference occurs in the photoresist layer. In terms of the AR layers previously discussed, the photoresist becomes the surrounding medium, refractive index n0, the new layer is the AR layer, refractive index nf, and the substrate remains as silicon, refractive index ns. Although the idea is conceptually simple, the thickness of the AR layer is rather difficult to determine. There are two main reasons for this. As the layer is interposed between the silicon and the photoresist the simple 107 The Production of Colour by Reflection

(a)

multiple photoresist reflections

silicon

(b)

rays λ /2 out of phase photoresist reflections cancel

antireflection layer

silicon

Figure 3.13 (a) Multiple light reflections in a film of photoresist on a silicon surface. (b) The deposition of an AR layer between the photoresist and the silicon results in cancellation of reflected beams by destructive interference, thus increasing the precision of the process

formula in Equation 3.3 cannot be applied and rather complex calculations of the reflectivity must be made. Second, the simple refractive index term n of the layer must be replaced by the complex refractive index N, because at wavelengths in the ultraviolet region many materials which are transparent at visible wavelengths absorb strongly. One suitable material that has been used in AR layers is silicon oxynitride, SiOxNy, often written as SiON. The compound has an advantage in that a change of composition alters the optical properties of the film (Table 3.1). The material is laid down as a thin film by passing a mixture of silane (SiH4), nitrous oxide (N2O) and nitrogen (N2) over the silicon wafer. The various proportions of the gases control the composition and, hence, allow the wavelength at which the film is optimally antireflective to bevaried at will. The SiOxNy layer is thus said to be a tuneable AR layer.

Table 3.1 Optical properties of Si–O–N ﬁlms at 248 nm wavelength as a function of composition

Composition Refractive index n Extinction coefﬁcient k

SiO0.86N0.24 1.8948 0.4558 SiO0.71N0.27 1.9682 0.5253 SiO0.54N0.59 2.0821 0.5004 SiO0.47N0.49 2.2127 0.6030 Colour and the Optical Properties of Materials 108

nf 1

nf 2

nf 3

nf 4

Figure 3.14 A surface foam can act as an AR coating. The reﬂectivity of the coating is determined by dividing up the surface into a large number of parallel layers and assessing the refractive index of each slice. The overall refractive index of the surface layers must equal the square root of the refractive index of the substrate for perfect AR behaviour

3.7.3 Graded index antireﬂection coatings

The problem offorming a single-film AR coating on a surface is contained in Equation 3.3. In the case of glass it has not proved possible to find a film material with a refractive index that fits this equation exactly. One solution to the problem is to make a film in which the refractive index varies gradually from that of the surrounding medium, usually air, to that of the substrate; that is, a GRIN material (Section 2.5). The first practical use of this idea, conceived more than 50 years ago, was to fashion a surface foam. The idea is to have a high concentration of air bubbles at the outer surface that gradually falls to zero at the inner (substrate) surface (Figure 3.14). Provided that the air bubbles are smaller than the wavelength of light, they are not resolved and the light encounters a medium in which the effective refractive index gradually increases from that of air to that of the substrate. Porous coatings of this type can be made from a silica gel. If a glass surface is dipped into the gel and then heated to form a porous glass layer, an antireflective surface coating can be formed which fits Equation 3.3 precisely. The refractive index of the film will depend upon the volume and size distribution of the pores, the polarisation of the incident light and will be wavelength dependent. To a first approximation, the assumption that the material behaves as a simple mixture (Section 2.5) can be employed. The average refractive index of the whole film is then given by Equation 2.11, which is also written in the form:

nf ¼ n0 þ Fðnsn0Þð3:4Þ where n0 is the refractive index of the surrounding medium, which also fills the pores, usually air, with n0 ¼ 1, and ns is the refractive index of the substrate; the amount of solid in the antireflecting layer is called the filling factor F (identical to the volume fraction of substrate Vs), which runs from 0 % at the surface to 100 % at the substrate. The idea of using GRIN optics in AR coatings, albeit in a slightly different form, was evolved in night-flying insects some millions of years ago. It is clearly of advantage to optimize the amount of light that the eyes of these night-flying insects receive and an AR coating on the eyes helps in this. The AR coating derives from the normal surface architecture of the insect eye. Adult insects use compound eyes, each of which is formed of many separate imaging units called ommatidia. The ommatidia form a hexagonal pattern of facets on the surface of the eye, each facet corresponding to the outer surface of an ommatidium. The eye facets of most day-flying insects, such as bees and dragonflies, are smooth, but certain night-flying moths and a few butterflies have facets that are covered with tiny bumps (Figure 3.15). The dimensions of the bumps are about half the wavelength of light, being about 200 nm at the base and 200 nm high. These bumps form an effective GRIN layer that endows the surface of each facet with marked AR properties. 109 The Production of Colour by Reflection

Figure 3.15 The antireﬂective GRIN structure on the surface of the eye of Morpho butterﬂy. [Reprinted by permission from Macmillan Publishers Ltd: NATURE, Photonic structures in biology, Pete Vukusic and J. Roy Sambles, 424, 852–855, copyright (2003)]

In order for the moth-eye surface to act as an AR layer, the protuberances must be small enough not to be resolved by light, which means in practice that the bumps must be separated by half the wavelength of the light or less. If this is not so, the array will act as a diffraction grating (Chapter 6). The surface architecture must also not be confused with surface roughness, which will increase diffuse reflection compared with specular reflection, but not decrease the total amount of light returned towards the source. The antireflectiveproperties of moth-eye surfaces can be determined by dividing the bumpy surface into slices parallel to the substrate surface, estimating the average refractive index and reflectivity of each layer and then summing over the slices to determine the reflectivity of the whole structure (Figure 3.16). This is a complex calculation, as the polarisation of the light must be taken into account. However, an approximate estimation of the effectiverefractiveindexcan be made using Equation 3.4. (An alternative approach via a diffraction problem is given in Section 6.2.) These bumpy surfaces are now being reproduced artificially for use as AR coatings. As the effect was first recorded on the surface of the eyes of some moths, these types of surface AR coating are called moth-eye AR coatings.The more formal name for a moth-eye AR coating is ultrahigh spatial-frequency surface relief grating. Nanoparticles can be used in a similar fashion. The refractive index of a layer of nanorods depends upon the constituents of the rods, the spacing between them and the angle at which they lie on the surface. The use of

nf 1

nf 2

nf 3

nf 4

Figure 3.16 A moth-eye surface structure can act as an AR coating. The reflectivity of the coating is determined by dividing up the surface structure into many thin layers and assessing the refractive index of each slice. The overall refractive index of the surface layers must equal the square root of the refractive index of the substrate for perfect AR behaviour Colour and the Optical Properties of Materials 110 several layers of rods can thus make a GRIN surface layer. The best AR coating (2009) has been made from five 100 nm layers of silica and titanium dioxide nanorods deposited at 45 to the surface of aluminium nitride plates, a material of use in LEDs and semiconductor lasers. The final layer, with an effective refractive index of 1.05, gives a reflectivity of 0.01 % (see this chapter’s Further Reading).

3.7.4 High-reﬂectivity surfaces

Thin-film coatings can also be used so as to optimize the reflectivity; that is, make thevalue of R as close to unity as possible. A film of thickness l/4 will achieve this provided that the refractive index of the film nf is greater than both n0 and ns. Two materials frequently used are SiOx,withx approximately equal to 1.0 (n 2.0), and TiO2 (n 2.5 2.8). ATiO2 film of thickness l/4 on glass will have a reflectivity of about 0.40 (40 %). As R for a single glass surface in air is about 0.04 (4 %), 40 % represents a tenfold improvement. The effect is used in costume jewellery. Rhinestones consist of a glass object with refractive index close to 1.52 coated with an approximately l/4 thickness film of TiO2. Variations in film thickness and viewing angle give these objects a wide variety of fleeting colours which are meant to simulate the fire of diamonds. Sparkling paints and nail varnish also make use of an approximately quarter-wavelength thickness of TiO2 deposited onto flakes of mica which are subsequently dispersed in the product. The various colours seen are created in a similar way to the colours on rhinestones.

3.7.5 Interference-modulated (IMOD) displays

Thin film interference is able to generate bright remarkably colours. This is exploited in a display technology aimed at mobile phone screens. The idea is based on the interference of white light falling on a pair of parallel reflecting surfaces, so that colour is essentially developed in a thin air film. The arrangement is similar to that of a Fabry Perot etalon . This device, which is an interferometer, consists of a semitransparent film separated from a fully reflecting film by a narrow air gap. Light from a broad source falling on the top surface is repeatedly reflected from the bottom surface and leaks from the top surface as a reflected beam and from the bottom surface as a transmitted beam. In a classic Fabry Perot etalon the transmitted beam is exploited and the reflected beam is suppressed. In an IMOD display the reflected beam is exploited. The arrangement of a single pixel consists of a pair of mirrors separated by a narrow air gap (Figure 3.17a). The principle of operation is as given in Sections 3.2 and 3.3. The pixel will reflect light of a wavelength l brightly for incident light falling normal to the surface when:

2d ¼ðm þ 1Þl 2 where d is the separation of the mirrors and mis an integer. As theviewer movesaway from normal incidence the colour will appear to move to shorter wavelengths; that is, red tends to move towards blue. The actual colour of the pixel will not be a singlewavelength, of course, but will depend upon the interference of thewhole spectrum, as described above (Figure 3.10). The device can operate in an interactive fashion if the separation of the air space between the mirrors is varied according to a controlled input. In current displays this is achieved by using electrostatic attraction between the top (fixed) and lower (moveable) film. Piezoelectric movement, used to control electrodes in a number of devices, including surface tunnelling microscopes, which are able to reveal atomic features on a surface, can also be utilized. As the separation varies, so the colour of the pixel changes (Figure 3.17b). These displays are currently being widely exploredfor mobile telephone screens. Theyhave an advantage in that although power is needed to change the colour of a pixel, once that colour is set, no power is needed to maintain it. In competing 111 The Production of Colour by Reflection

daylight

semi-transparent mirror air

(a) mirror

daylight

no reflection (black)

(b) Figure 3.17 The principle of operation of an IMOD display: (a) reﬂection of light from a pair of parallel mirrors (a Fabry–Perot etalon); (b) variation of mirror separation gives rise to different coloured pixels technologies, such as liquid crystal displays or organic light emitting diode displays, the power must be supplied continuously to maintain colour and brightness. Moreover, these pixels are easily visible in bright daylight, which is a drawback of some present displays.

3.8 Multiple Thin Films

3.8.1 Dielectric mirrors

Traditionally, mirrors have been made from metals. The best metallic mirrors are made of a thick layer of silver, which has a reflectivity of about 0.96 in the visible. (The reflectivity of metals is considered in more detail in Section 10.15.) Surprisingly, multiple thin films of transparent materials can be laid down one on top of the other in such a way as to form perfect mirrors. These are often called dielectric mirrors. The fabrication of such devices forms part of the subject area known as photonic or thin-film engineering. Awide variety of multilayer mirrors are now manufactured, mainly from oxides and fluorides. These are all stable in air and have the additional advantage over metallic mirrors of not degrading in normal use. The simplest formula for the reflectance of such a mirror refers to the specific case in which all layers are l/4 thick and of alternating high (H) and low (L) refractive indices, nH and nL, illuminated by light falling perpendicular to the surface. The arrangement (Figure 3.18) is called a quarter-wave stack. The maximum reflectance of a quarter-wave stack deposited on a substrate in the sequence:

substrate; L; H; L; H; L; H;...L; H; air Colour and the Optical Properties of Materials 112

n0 (air)

λ/4 nH 1 pair λ/4 nL

ns (glass)

Figure 3.18 A stack of thin ﬁlms, each of optical thickness l/4, called a quarter-wave stack, can act as an effective dielectric mirror. The reﬂectivity increases with the number of pairs of layers and rapidly approaches 1.0 is given by the formula: 2N 2 ¼ ns f n0 R 2N ns f þ n0 where f is equal to (nH/nL), n0 is the refractive index of the surrounding medium, usually air (n0 ¼ 1.0), ns is the refractive index of the substrate, usually glass (ns 1.5) and N is the number of (LH) pairs of layers in the stack. For a stack in air this equation is equivalent to: "# 2 n f 2N 1 2 n ðn =n Þ2N R ¼ s ¼ s L H 2N þ 2N ns f 1 ns þðnL=nHÞ

Computation shows that, as the number of pairs of layers increases, R rapidly approaches 1.0, implying perfect reflectivity. The form of the reflectivity as a function of wavelength for light falling on the stack at normal incidence has a typical structure consisting of a central plateau together with small side maxima distributed about the design wavelength l0 (Figure 3.19). In general, the central plateau becomes squarer and higher as the number of layers increases until a reflectivity of unity is reached. The width of the central plateau is given by: 4l 1f Dl ¼ arcsin p 1 þ f where f ¼ (nH/nL). Different formulae must be used if the stack terminates with an L-layer, if there are not complete sets of pairs of layers or for oblique illumination. When the beam is at an oblique angle of incidence the polarisation of the beam must also be taken into account. 113 The Production of Colour by Reflection Reflectance

λ 0 Wavelength Figure 3.19 General form of the reﬂection from a quarter-wave stack as a function of the wavelength for light at normal incidence

3.8.2 Multilayer stacks

The difficulty of making calculations of the reflectivity and transmissivity of thin-film multilayers prevented large-scale use of this technology in the first half of the twentieth century. The mathematical formulation of the problem, though, was solved at this time, and the optical properties of a stack can be described by a methodology that allows the contribution of each layer to be represented by a matrix. The total optical effect of the stack is obtained by multiplying the matrices together. Suitable computational software is now readily available (see this chapter’s Further Reading). The general approach used to make a multilayer optical component is to lay down a stack of thin films which have alternately higher and lower refractive indices using vacuum evaporation of the materials. Manipulation of the thicknesses and the refractive indices of the layers in the stack, in accordance with computation, allows for the modification of the optical properties at will. This technology is thus equally suitable for the production of multilayered AR coatings. For example, Figure 3.20a d shows the variation in reflectivity of a stack of four thin films as the thickness of just one of the layers is changed. The four thin films are deposited on a glass substrate and alternate between high refractive index (H) and low refractive index (L), ending with air. The arrangement of the layers is: air (n ¼ 1.0); (1) L, 93 nm, n ¼ 1.48; (2) H, 120 nm, n ¼ 2.30; (3) L, 37 nm, n ¼ 1.48; (4) H, variable thickness, 30 nm, 24 nm, 18 nm, 12 nm, n ¼ 2.30; substrate, glass, n ¼ 1.52. The single layer to be changed was that next to the glass substrate, and then only from a thickness of 30 nm to 12 nm. The curves are all evaluated for a design wavelength of 550 nm and for light normally incident upon the stack. The final curve (Figure 3.20d) makes an almost optimal AR coating. In general, when a multilayer stack is tilted the reflectivity must take into account polarisation. Although this has only a small effect on the total reflectivity, the wavelength which is strongly reflected or transmitted shifts towards lower values, butit does so much more slowly than for a single thin film. Thus, the film will look bluer as the stack is tilted. It is found that the central wavelength will decrease from l0 for normal incidence to l when the stack is tilted through small angles (than 20) given by the expression:

l ¼ l ½ ð2= 2Þ 1 2nf Colour and the Optical Properties of Materials 114

7 a 6

4 b 3 Reflectance / % 2 c 1 d 0 450 500 550 600 650 700 750 Wavelength / nm Figure 3.20 The reﬂectivity of a stack of four thin ﬁlms on a glass substrate in air. The thickness of each layer is constant except for the one adjacent to the glass, which takes values of (a) 30 nm, (b) 24 nm, (c) 18 nm, (d) 12 nm. The stack (d) shows almost perfect AR behaviour. Computations were made using ‘Filmstar’ software (see this chapter’s Further Reading)

where nf is the effective refractive index of the stack and is in radians. This means that these multilayer stacks are tunable over small degrees of rotation. For accurate work the stack must be aligned precisely to ensure wavelength-specific performance. If the layers are uneven in thickness, or to some extent disordered, a wide variety of wavelengths will be reflected. These will be perceived as white or silver, depending upon the smoothness of the surfaces. This is the reason why a roll of thin plastic film used for food wrap looks silver. Many insects show silver markings that are similarly made up of thin layers of transparent material of varying spacings (Figure 3.21).

3.8.3 Interference ﬁlters and distributed Bragg reﬂectors

The same technique of multiple dielectric layer deposition can be used to make interference filters. The form of the reflection curve of a multilayer stack (Figure 3.19) shows that wavelengths to either side of the central plateau will be transmitted and those within the plateau will be reflected. By using multiple thin films the regions that transmit or reflect can be precisely manipulated to make optical filters. These fall into three different categories. Shortpass filters transmit visible wavelengths and cut out infrared radiation (Figure 3.22a). They are often used in surveillance cameras to eliminate heat radiation. Longpass filters block ultraviolet radiation and transmit the visible (Figure 3.22b). Other filters, called bandpass filters, pass only a limited section (or band) of the electromagnetic spectrum (Figure 3.22c). (These thin-film interference filters generally give a much sharper transmittance than the type of filter made from dye molecules distributed in a gelatine matrix; the type of filter illustrated in Figure 1.18). As the filters are made of transparent layers, the wavelengths not transmitted are reflected. Bandpass filters, therefore, act as mirrors for the complementary colour of the transmitted band. Because of this effect, these filters are often vividly coloured (Figure 3.23). When multilayer reflectors are included in an optical device such as a waveguide or some types of laser they are called distributed Bragg reflectors. They are typically made from layers of TiO2 and SiO2. The reflectivity of such a multilayer is computed in the same way as any multilayer stack, taking into account the surroundings 115 The Production of Colour by Reflection

Figure 3.21 The Silver-washed Fritillary butterfly, Argynnis paphia. The silver ‘wash’ on the wings is caused by reflection from a disordered thin-film multilayer stacks and the substrate on which the Bragg reflector is deposited. The wavelength interval that is reflected from such a reflector is called the photonic stopband. The approximate width of the stopband is given by the formula: 4l 1f Dl ¼ arcsin p 1 þ f where f ¼ nH/nL.

3.9 Fibre Bragg Gratings

Multilayer interference filters can also be produced in the cores of optical fibres. These are called fibre Bragg gratings (FBGs) and are used for controlling light signals as they travel along the fibre. FBGs are formed within the core of an optical fibre (usually a monomode fibre) (Section 2.9) in which the refractive index is modulated in a periodic way with a repeat spacing d of about the wavelength of light. The formation of an FBG was first observed in 1978, more or less by accident, rather like the initial observation of the formation of frequency-doubled light in a fibre (Section 4.11). Light from an argon-ion laser was focused into a length of germanium dioxide (GeO2)-doped silica (SiO2) fibre and, surprisingly, more and more light was reflected back along the fibre as time passed. It was concluded that a refractive index grating was being created in the fibre by interference between the incident wave and a wave reflected from the far end of the fibre. The two waves formed an interference pattern in the fibre, which produced the refractive index change. Although initially treated as a bizarre phenomenon, it has since been found that any GeO2-doped SiO2 fibrewill behave in a similar fashion. The refractive index gratings that form in this way are called Hill gratings.Hill gratings are limited to the wavelength of the radiation producing the effect. Colour and the Optical Properties of Materials 116

(a) 100

50 % Transmission

400 700 1000 Wavelength / nm

(b) 100

50 % Transmission

400 700 Wavelength / nm

50 % Transmission

400 700 1000 Wavelength / nm

Figure 3.22 Transmission profiles of multilayer dielectric filters: (a) a shortpass filter; (b) a longpass filter; (c) a bandpass filter 117 The Production of Colour by Reflection

Figure 3.23 Multilayer interference filters. The bright reflected colours are complementary to the colours transmitted by the filters and absorbed by the black backing

The simplest (conceptual) modulation is a step repeat (Figure 3.24). This type of grating is described as a uniform grating. If the spacing of the refractive index modulation is not constant, but varies in spacing in a uniform way along the length of the modulation from d1 to d2, the grating is said to be chirped. All can be described as a form of distributed Bragg reflector (Section 3.8). Theway in which FBGs influence light pulses passing down the core of the fibre can be understood in terms of multiple thin-film optics. The light will be reflected back from the grating if thewavelength of the light l and the spacing of the grating d are given by:

l ¼ 2navd

(a) n1 cladding

n2 n3 core

(b) Refractive index

n2 Position d Figure 3.24 FBGs. (a) Periodic modulation of the refractive index in the core of an optical ﬁbre. (b) The simplest step modulation of refractive index. The refractive indices of the cladding, core and modiﬁed core are n1,n2 and n3 respectively Colour and the Optical Properties of Materials 118

where nav is the average of the refractive index of the core and the modulated region. The value of the refractive index change Dn between the modulated and unmodulated regions is of the order of 10 3 to 10 4 on the core refractive index, which is close to 1.5. It is clear, therefore, that a single modulation would hardly cause any change in a pulse. However, FBGs are often of the order of 40 000 modulations in length (which only occupies 200 nm of fibre for blue green light of 500 nm wavelength) and, hence, considerable intensity can be reflected. A pulse of white light introduced into a fibre containing such a grating would reflect back a pulse of monochromatic blue green light of 500 nm wavelength. The other wavelengths would be transmitted. The grating is thus able to act as a filter or as a mirror, as in the case of the other multiple thin-film devices. There are a number of ways of fabricating FBGs. The simplest is to use the interference of two ultraviolet laser beams shone onto the fibre from the side. The peaks and troughs of the interference pattern of two beams focused on the fibre create the refractive index changes required (Figure 3.25a). Because the spatial frequency of the interference pattern is readily changed by altering the angle at which the beams meet, a wide range in the spacing of the refractive index modulations can be imposed upon the core. Gratings can also be created by using a mask, which, because of the dimensions involved, acts as a diffraction grating to create a pattern of maxima and minima in the fibre core (Figure 3.25b). If the part of the fibre which lies in the interference pattern is bent into a curve, chirped gratings can be made. As described above, FBGs form under the influence of external radiation, most often of ultraviolet frequencies. However, not all fibres are susceptible to the formation of refractive index gratings. The glass must be photosensitive; that is, they react to light in a specified way (see also Sections 10.17 and 10.18). Although GeO2-doped SiO2 glass is satisfactory, much better gratings form in fibres which also contain boron

(a)

ultraviolet ultraviolet light light

(b) ultraviolet light

mask

Figure 3.25 Fabrication of FBGs: (a) interference of two beams of ultraviolet light; (b) diffraction pattern from a mask 119 The Production of Colour by Reﬂection

dropped added from signal to signal Figure 3.26 The addition and removal of a signal from a ﬁbre using an FBG; schematic

trioxide (B2O3) or tin dioxide (SnO2) as co-dopants. Additionally, fibres can be transformed into a photosensitive state by forcing hydrogen into the structure. There is some uncertainty about the mechanism by which the change in refractive index is produced. It is agreed, though, that defects in the structure are involved in some way. From a number of possibilities, local density variation, the formation of colour centres involving GeO or the formation of centres involving a germanium hydrogen (GeH) pair seem to be the most likely candidates at present. There is considerable interest in FBGs because they have numerous applications in fibre-optic communications. Clearly, each different wavelength that passes down a cable can carry data. As colour signals do not become mixed, the more wavelengths that can be crammed into a fibre the more data that it can carry per unit time. The technique of putting large numbers of different wavelengths down a fibre is called dense wavelength division multiplexing (DWDM). In this context, FBGs can be used for adding or removing signals from a fibre, necessary in wavelength multiplexing of optical communication systems (Figure 3.26).

3.10 ‘Smart’ Windows

Smart windows are those that respond to changes in the external and internal environment. There are a number of different types under active investigation. Here, just two are mentioned, both of which rely on thin-ﬁlm reﬂectivity for the active function, low-emissivity windows and self-cleaning windows. For other approaches see Section 10.12.

3.10.1 Low-emissivity windows

Windows in buildings are targets for improved energy efficiency. The reason for this is that normal window glass is an extremely good absorber and emitter of thermal energy. The black-body equations (Section 1.6) show that a room with a temperature of 21 C has approximately 94 % of the thermal energy in the range 5 40 mm, with a peak at about 10 mm. Glass absorbs and re-emits about 80 % of this energy, making windows an appreciable gateway for loss of heat. Windows which address this problem are known as low-emissivity windows. The details depend upon the place of use. In colder regions it is not only necessary to minimize heat loss to the outside, but also to guarantee that solar energy penetrates the glass and acts as a passive heating agent. In desert regions it might be more desirable to make reflection of external solar energy the priority. All the systems in use rely on coating the inside of one or both panes of a double sheet of glass with a thin film of material which, in simple terms, is transparent tovisiblewavelengths and opaque to infrared. The positioning of the coating depends upon the use for which thewindow is designed. To prevent heat loss from rooms in cooler climates the coating is frequently upon the inside of the inner pane (Figure 3.27a). One commonly used substance is tin dioxide (SnO2) doped with fluoride ions (F ), with a refractive index of approximately 2.0. This material is transparent to visible wavelengths but strongly absorbent to the wavelengths that characterize the thermal energy from the room, thus capturing the energy. These films have a Colour and the Optical Properties of Materials 120

(a)

outside room

glass glass air gap n ≈ 1.5 n ≈ 1.5 - SnO2/F n ≈ 2.0

(b)

outside room

colour suppression layer, n ≈ 1.75

Figure 3.27 Low-emissivity coatings: (a) the coating is applied to the inside of a double glass unit and on the side nearest to the room; (b) another thin film with a lower refractive index is often applied to reduce reflections and act as a colour suppression layer low emissivity for these wavelengths and, hence, cannot lose the energy by radiation. The energy is conducted back through the glass and returned into the room by radiation from the uncoated surfaces. The useful performance of the film is limited by its thickness. As films become thicker, the emissivity increases, so it is important to keep film thickness low. Unfortunately, the ideal thickness for SnO2 films is exactly that which produces a green colour due to interference and gives green-tinted windows. Thegreen reflection from the doped SnO2 layer can be suppressed by coating the glass with a thin layer of transparent material with a lower refractive index than the SnO2 before the SnO2 is applied (Figure 3.27b). Colour suppression occurs via the same principles outlined in Section 3.7.2. The aim is tocause reflections from the top and bottom surfaces ofthe doped SnO2 layerto be out of phase and so interfere destructively, hence eliminating colour production via interference. If the film is viewed from inside the room, to a first approximation it is convenient to call the glass pane the surrounding medium, refractive index n0, and the doped SnO2 coating as the substrate. Using the formula for a l/4 AR layer, Equation 3.2, shows that the ideal film refractive index nf is given by: p nf ¼ n0ns ð3:5Þ where n0 is the refractive index of the doped SnO2 layer and ns is the refractive index of the glass (1.5). This suggests that a thin film with a refractive index which is of the order of 1.75 might form a suitable colour 121 The Production of Colour by Reflection suppression layer for someone inside the room. Exact multiple thin-film calculations are needed to refine the thin-film characteristics, both in reflection and transmission.

3.10.2 Self-cleaning windows

Self-cleaning windows need to destroy organic molecules and bacteria that stick to the glass. An additional desirable property is that the surface should be hydrophilic so that water flows over it readily and allows debris to be washed away with rain. Titanium dioxide (TiO2) is an oxidation photocatalyst. It is able to decompose organic molecules and disrupt the surfaces of bacteria when irradiated with ultraviolet light. This has made it a promising surface coating for ‘self-cleaning’ widows, as normal sunlight contains sufficient ultraviolet to effect the removal of organic deposits on window surfaces over the course of the day. Moreover, thin films of TiO2 pick up hydroxyl (OH ) groups on the surface, making it hydrophilic. Thus, self-cleaning windows use external coatings of TiO2. As withlow-emissivitywindows,the thinfilmcausesunwantedinterference effects. Inthiscase,thepresence of a l/4 thin film of TiO2 on the window increases surface reflectivity greatly. This is given by Equation (3.2): n2n n 2 R ¼ f 0 s ð3:2Þ 2 þ nf n0ns where nf is the refractiveindex of the TiO2 film, n0 is the refractive indexof air (1.0) and ns the refractiveindex of the window glass (1.5). There are two common forms of TiO2: anatase, with an effective refractive index in thin film form of approximately 2.52, and rutile, with an effective refractive index in thin film form of approximately 2.76. Substituting these into Equation 3.2 shows that the reflectivity of the surface will lie between approximate values of 38 and 45 %. Both of these are too high for convenient use in ordinary windows. It is possible to try to suppress this high reflectivity by the inclusion of an AR coating between the TiO2 film and the glass. However, this faces the same problem as described above for low-emissivity windows, and it is not easy to find a film that suppresses high reflectivity when viewed from both sides of the glass. GRIN techniques can help. Self-cleaning windows fabricated with a surface coating of porous silica about 120 nm thick containing nanoparticles of TiO2 are able to combine both the self-cleaning and AR properties in one. As described above, a l/4 AR layer on glass should ideally possess a refractiveindexof about 1.225 (Section 3.7.1). Porous silica can give a lower value than this, which is increased by the presence of the TiO2 nanoparticles. The refractive index of the film can be calculated using the methods in Section 2.5, that is:

nf ¼ n1V1 þ n2V2 þ n3V3 þ where n1 represents the refractive index of component 1, etc. and V1 represents the volume fraction of the material 1, etc.:

V1 þ V2 þ V3 þ ¼1

In practice one would use computer software to evaluate the ideal thicknesses of the TiO2 and SiO2 (or other) AR layers so as to optimize the transparency of the window.

3.11 Photonic Engineering in Nature

The application of multiple thin films in nature is widespread, and a volume could easily be written on this topic alone. If the film thicknesses are fairly uniform, then a bright colour will be reflected. Such colours are generally Colour and the Optical Properties of Materials 122 referred to as iridescent, meaning that the colour has a metallic appearance and the tone changes with viewing angle. If the layers are uneven in thickness, or to some extent disordered, a wide variety of wavelengths will be reflected. These will be perceived as white or silver, depending upon the smoothness of the surfaces (see Figure 3.21). All of these are referred to as structural colours to differentiate them from colours produced by pigments. Here, just a few examples from a legion are touched upon. Many more will be found if the references (see this chapter’s Further Reading) are consulted.

3.11.1 The colour of blue butterﬂies

An example of the vivid blue colouring seen in butterflies is provided by the Common Blue, Polyommatus icarus (Figure 3.28a). The colour arises in tiny scales that cloak the wings (Figure 3.28b). The colour perceived is built up by a mosaic of these tiny scales and is similar in result to that used by pointillist painters such as Seurat. The blue scales of this butterfly are made up of sheets of transparent multilayers running parallel to the scale base (Figure 3.29a). There are four layers of transparent material with a thickness of about 50 nm and a refractive index of about 1.57 separated by air layers of approximately twice this dimension. In addition, the layers of transparent material are be perforated into a ‘pepper-pot’ structure (Figure 3.29b). Calculation confirms that this arrangement is highly reflective for violet blue wavelengths. In nature, there are many similar species of blue butterfly, each of which is characterized by a different tone of blue and which can be recognized one from another by these subtle differences. It is easy to appreciate that the colour of the reflective scales can be tuned by small changes in the multilayer thickness, spacing and degree of perforation. This latter attribute is equivalent to a GRIN layer that has a refractive index somewhere between that of air and 1.57.

3.11.2 Shells

Many shells have a multilayer construction, as this affords the desirable combination of strength and lightness. Occasionally this feature gives rise to structural iridescent colours. The colours are more often visible on the inside of a shell, as the outsides tend to be camouﬂaged or otherwise coloured to aid concealment. In many species these colours are pale greens and pinks and are known as mother-of-pearl or nacre. However, the New Zealand paua, Haliotis iris, has a very marked iridescence and displays intense colours that change with viewing direction (Figure 3.30). The colours exhibited are many vivid blues and greens. The multilayers giving rise to this spectacular effect are derived from alternating organic and inorganic layers. The colour effect is enhanced by dark-pigmented underlying material that absorbs any light that has not been reﬂected. The shells are used for decoration and jewellery.

3.11.3 Labradorite

Minerals can develop as multilayer structures in a number of ways. An example is provided by the mineral labradorite. This material exhibits ﬂashing rainbow-like colours which vary as the angle of observation changes in a typically iridescent fashion (Figure 3.31). The phenomenon is also known as schiller and labradorescence when applied to the mineral. Most commonly the colours exhibited are violets and blues, but greens, yellow and orange colours can also be seen in some specimens. Geologically, labradorite is a plagioclase feldspar; feldspars being minerals constructed from a strong 4 framework of corner-sharing (SiO4) groups with alkali or alkaline earth cations contained in the cages present. It has a composition lying between the parent compounds anorthite (CaAl2Si2O8) and albite (NaAlSi3O8), both of which are also feldspars. Labradorite consists of between 50 % and 70 % anorthite, so that its formula can be written as Ca0.5–0.7Na0.5–0.3(Al,Si)AlSi2O8. 123 The Production of Colour by Reﬂection

Figure 3.28 (a) The Common Blue butterﬂy P. icarus. (b) Scales from the wing of P. icarus. Only some scales have a blue reﬂecting microstructure. The yellow–brown scales are coloured by melanin-related pigments. [Figure (a) reproduced with kind permission of Dr J.A. Findlay] Colour and the Optical Properties of Materials 124

Figure 3.29 Electron micrographs of a blue scale from the wing of the butterﬂy P. icarus. (a) Transmission electron microscope transverse section. (b) Scanning electron micrograph of a fractured blue scale. The multiple internal layers with a perforated structure that give rise to the blue reﬂectivity are revealed

It is believed that, during the formation of the parent rocks, the feldspar leading to labradorite had a homogeneous composition in which the various cations were distributed at random over the possible sites available. This is known to happen at high temperatures, and a complete solid solution is said to form between the parent phases anorthite and albite. However, at low temperatures this homogeneous solid is thermody- namically unstable and over geological time scales the sodium and calcium ions segregate to form alternating lamellae which are sodium rich and calcium rich. This also necessitates the diffusion and subsequent ordering of aluminium and silicon cations at the same time. The result is that adjacent layers possess differing refractive indices. In rare circumstances the segregation can result in lamellae which have the appropriate thickness and degree of ordering to reflect visible light and a multiple thin-film structure results. For example, an investigation of the microstructures of labradorite giving rise to a blue schiller had stacks of alternating lamellae of dimensions 72.5 nm and 65.1 nm, whereas materials showing a red schiller had lamellae of 176.6 nm alternating with lamellae of thickness 87.4 nm. As expected, the colours observed will depend upon the relative thickness of the lamellae and the angle of illumination and observation, and the refractive indices of the component lamellae. As these are subject to many variables, no two samples of labradorite from different locations are truly identical. 125 The Production of Colour by Reflection

Figure 3.30 New Zealand paua (H. iris) shells showing characteristic iridescent colours that are noticeably angle dependent

3.11.4 Mirror eyes

The design of eye most familiar to readers is the ‘camera’ type, which uses a lens to focus light onto a light- sensitive membrane the retina. However, a number of different eye designs are found in nature, some of which use mirrors rather than lenses. Just one from many examples is provided by scallops of the genus Pecten, which

Figure 3.31 A specimen of labradorite from Madagascar. The colours displayed (labradoresence or schiller) change with viewing angle Colour and the Optical Properties of Materials 126 have focusing elements made up from multilayers of cytoplasm, with a refractive index of 1.34, and guanine crystals, with a refractive index of 1.83. These layers form a mirror to bring light to a focus. The total thickness of the mirror is about 6 mm and contains 60 or so layers. The multilayer mirror is hemispherical in shape and forms the interior rear surface of the eye so that rays of light entering the eye fall more or less perpendicularly upon the stack. As a rough estimate, each layer has an optical thickness of about l/4, corresponding to strong reﬂection of l ¼ 600 nm. However, the layers of the mirror are not evenly spaced, and for this reason the eyewill focus a range of wavelengths.

Appendix A3.1 The Colour of a Thin Film in White Light

a b c Retardation /nm Colour reflected Colour transmitted Start of first order 0 black bright white 40 iron grey white 97 lavender grey yellowish white 158 grey blue brownish white 218 grey brownish yellow 234 green white brown 259 white bright red 267 yellow white carmine red 281 straw yellow deep violet 306 bright yellow indigo 332 yellow blue 430 yellow brown grey blue 505 orange red blue green 536 red green 551 deep red yellow green 555 End of first order; start of second order 565 magenta purple bright green 575 violet green yellow 589 indigo gold 609 dark blue yellow 664 sky blue orange 680 blue orange brown 728 blue green brown orange 747 green carmine red 826 bright green purple red 843 yellow green violet purple 866 green yellow violet 910 yellow indigo 948 orange dark blue 998 orange red green blue 1050 crimson violet yellow green 1100 dark violet red green 127 The Production of Colour by Reflection

Appendix A3.1 (Continued)

a b c Retardation /nm Colour reflected Colour transmitted 1110 End of second order; start of third order 1128 blue violet yellow green 1151 indigo off yellow 1258 blue green pink 1314 emerald green red 1334 sea green brown red 1350 green purple violet 1376 dull green violet 1400 yellow green violet grey 1426 green yellow grey blue 1450 yellow indigo 1495 rose pink sea green 1534 carmine red green 1621 dull purple dull sea green 1650 violet grey yellow green 1665 End of third order; start of fourth order 1682 blue grey green yellow 1710 dull sea green yellow grey 1750 blue green lilac 1800 green brown purple red 1811 green carmine 1900 pale green red 1927 greenish grey grey red 2000 pale grey blue grey 2100 carmine red green 2220 End of fourth order; start of fifth order 2500 green 2700 pink Beyond this point, orders overlap and the film colour is generally pale pink or pale green in reflection. a The retardation is equal to the path difference p between the interfering rays. For a single film, p ¼ 2nd, where n is the refractive index of the film and d p ¼ d n n d n n (nm) is the physical thickness. For a birefringent crystal, (| 1 2|), where is the thickness of the slice of crystal and 1 and 2 are the effective d n n refractive indices of the slice for light of two perpendicular polarisation directions. For a uniaxial crystal this is maximally (| 0 e|) (see Chapter 4). b This colour is seen in reflection from a thin film in air when illuminated by white light at normal incidence. It is the same colour as that shown in transmission by a thin transparent plate of an anisotropic crystal viewed at normal incidence in white light between crossed polars (see Chapter 4). c This colour is the complementary colour to that reflected and is the same as that shown in transmission by a thin film in air when illuminated by white light at normal incidence. It is the same as that shown in transmission by a thin transparent plate of an anisotropic crystal viewed in white light between parallel polars (see Chapter 4). In addition, these colours are seen in reflection when a thin transparent film on a substrate with a greater refractive index is viewed at normal incidence in white light. Further Reading

Much of this chapter is concerned with thin-ﬁlm optical engineering. An introduction to the topic is given by E. Hecht, Optics, 4th edition, Addison-Wesley, San Francisco, 2002. B. E. E. Saleh, M. C. Teich, Fundamentals of Photonics, John Wiley and Sons, Inc., New York, 1991. Colour and the Optical Properties of Materials 128

An interesting historical perspective of the evolution of CDs and DVDs is A. E. Bell, Sci. Am. 275 (July), 28 32 (1996). R. L. Gunshor, A. V. Nurmikko, Sci. Am. 275 (July), 34 37 (1996). G. Zorpette, Sci. Am. 283 (August), 19 20 (2000). S. Nakamura, M. Riordan, Sci. Am. 300 (April), 54 59 (2009). The colours produced by soap films are described and explained in C. Isenberg, The Science of Soap Films and Soap Bubbles, Tieto, Clevedon (UK), 1978. Reprinted by Dover, New York, 1992. The use of nanorods as an antireflective surface is given in J.-Q. Xi et al., Nat. Photonics 1, 176 179 (2007). Complete coverage of the theory of single and multiple thin films is in H. A. McLeod, Thin-Film Optical Filters, 3rd edition, Institute of Physics, London, 2001. For background on the history of thin-film computation, see O. S. Heavens, Rev. Prog. Phys. 23, 1 65 (1960). For IMOD displays, see M. M. Waldrop, Sci. Am. 297 (November), 68 71 (2007). Free versions of the software ‘Filmstar’, for the computation of thin-film optics (and used to compute Figure 3.20), are available from Dr F. T. Goldstein, FTG Software Associates, PO Box 597, Princeton, NJ 08542, USA (www.ftgsoftware.com). Full information on FBGs will be found in R. Kashyap, Fiber Bragg Gratings, Academic Press, London, 1999. The topic of colour in nature is described from an evolutionary perspective, with examples of thin-film colours, mirror eyes, etc. by A. R. Parker, In the Blink of an Eye, Free Press, London, 2003. Structural colour is reviewed by P.Vukusic, Structural color, in Dekker Encyclopedia of Nanoscience and Nanotechnology, J. A. Schwarz, C. I. Contescu, K. Putyera, (eds), Vol. 5, Marcel Dekker, New York, 2004, pp. 3713 3722. Many aspects of structural colour, including butterfly scales, moth-eye AR surfaces and mirror eyes, will be found in the following papers and the references cited therein: P. Vukusic, R. J. Wooton, J. R. Sambles, Proc. R. Soc. Lond. Ser. B 271, 595 601 (2004). P. Vukusic, J. R. Sambles, C. R. Lawrence, R. J. Wooton, Proc. R. Soc. Lond. Ser. B 269, 7 14 (2002). A. R. Parker, D. R. McKenzie, M. C. J. Large, J. Exp. Biol. 201, 1307 1303 (1998). A. R. Parker, Z. Hegedus, R. A. Watts, Proc. Roy. Soc. Lond. Ser. B 265, 811 815 (1998). A. R. Parker, Am. Sci. 87, 248 255, (1999). H. Ghiradella, Appl. Opt. 30, 3492 3500 (1991). D.-E. Nilsson, Nature 332, 76 78 (1988). A. A. Fincham, Nature 287, 729 731 (1980). M. F. Land, Sci. Am. 239 (December), 88 99 (1978). 4 Polarisation and Crystals

. Why do some crystals produce double images? . How can infrared radiation be changed into green light? . How do liquid crystal displays form images?

The interaction of crystals and light has long produced fascinating and puzzling experimental results. In the preceding chapters, the polarisation of light has not ﬁgured prominently. However, when the interplay of light and crystal symmetry is considered the polarisation of light can no longer be ignored. This chapter describes these and related interactions and shows how they lead to colour generation.

4.1 Polarisation of Light

Light can be regarded as a wave of wavelength l with electrical and magnetic components lying at right angles to one another, each described by a vector (Chapter 1). For the majority of optical processes only the electric vector E is important and the light can be represented by a sinusoidal wave that describes the amplitude of E as a function of position and time (Figure 4.1a). The vector E is always perpendicular to the direction of propagation of the light but can adopt anyangle otherwise, similar to the positions available to a hand on a clock. For ordinary light, such as that from the sun, the orientation of the electric vector changes in a random fashion every 10 8 s or so, as if the seconds-hand of a clock jumped unpredictably from position to position without rotating in a steady manner. The position of the electric vector deﬁnes the polarisation of the light wave. Ordinary light is said to be unpolarised.

Colour and the Optical Properties of Materials Richard J. D. Tilley 2011 John Wiley & Sons, Ltd Colour and the Optical Properties of Materials 130

(a) y

L z R

electric field vector E

(b) y (c) y

θ x x

(d) y (e) y

θ x x

Figure 4.1 (a) A snapshot of a light wave moving from left to right with the electric field vector E in the plane of the paper at that instant. When viewed along the ray the tip of the electric field vector may: (b) oscillate along a line at a constant angle u to form a linearly polarised beam; (c) trace out an ellipse to form an elliptically polarised beam; (d) trace out a circle to form a circularly polarised beam. The position of the electric field vector is shown at three different instants in (c) and (d). (e) The electric field vector at any instant can be resolved into two mutually perpendicular components

Light is described as linearly or plane polarised when the electric vector E which describes the light wave is forced to vibrate in a single plane.1 This is analogous to the seconds-hand on a clock being stuck permanently in one position. The plane that constrains the electric ﬁeld vector can lie at any angle to the propagation direction

1 In this book, the shorthand term ‘polarised light’ will be taken to mean linearly polarised light. Other forms of polarisation will be described explicitly. 131 Polarisation and Crystals

(Figure 4.1b). On looking into the beam, if the tip of the E vector were visible, it would appear to oscillate along a fixed line at an angle . In the most general representation of polarised light, the tip of the vector E traces out an ellipse, called the vibration ellipse, and the light is said to be elliptically polarised (Figure 4.1c). If the tip of the electric field vector traces out a circle the light is said to be circularly polarised (Figure 4.1d). Taking time into account, the true path of the tip of the vector forms a helix in both cases. Elliptically or circularly polarised light can be right polarised, so that the tip of the E vector traces out a path like that of a screw thread a right-handed helix. Left circularly polarised light has the tip of the E vector travelling in the opposite way as a left-handed helix. Both circularly and linearly polarised light are special forms of elliptically polarised light, in the first case when the major and minor axes of the ellipse are equal and in the second case when one of the axes is zero in length. A linearly polarised light beam can also be considered to be composed of a superposition of right and left circularly polarised light beams with the same frequency and amplitude. This approach is useful in the discussion of optical activity (Section 4.12). It is often convenient, when discussing the effects of polarised light on a material, to resolve the electric field vector into two components along (any) mutually perpendicular axes, x and y (Figure 4.1e). The amplitude of the component directed along the y-axis varies from a maximum value þ y to a minimum value y through zero, while the amplitude of the component directed along the x-axis varies from a maximum value of þ x to a minimum value of x through zero. Each resolved component is linearly polarised. These two parts may or may not be in phase with each other initially; that is, the maximum þ y may or may not coincide with the maximum þ x value. After interaction with a nonisotropic material, the phase between the two components will be changed. The resultant beam can be obtained by recombination of the two components in the reverse procedure to that used to divide the initial beam. The shape of the vibration ellipse depends upon the amplitudes of the x- and y-components and the phase difference between them. Thus, a general beam of polarised light will emerge from an isotropic material, such as a glass or a cubic crystal, with unchanged polarisation but from a nonisotropic material with a different polarisation. Many light beams can be considered to be composed of two fractions, one polarised and one unpolarised. The relative amount of each is expressed as the degree of polarisation of the light. While unpolarised light can interfere freely, it is important to note that two light beams polarised perpendicular to one another do not interfere or form interference patterns. In the case of linearly polarised light, it is often helpful to resolve the polarisation into two components perpendicular and parallel to the plane of incidence of the beam the plane which contains the incident ray, the normal to the surface and the reflected ray. The component of the light wave polarised such that the electric field vector lies in the plane of incidence is called the p-wave,ortransverse magnetic (TM) wave. The component of the light wave polarised such that the electric field vector lies perpendicular to the plane of incidence is called the s-wave,ortransverse electric (TE) wave (Figure 4.2). In general, s- and p-waves differ in the way they are reflected and refracted. Whilst the human eye is unable to detect polarisation direction, many animals have this ability. A number of examples will be given later in this book.

4.2 Polarisation by Reﬂection

When light is incident upon the surface of a transparent dielectric such as glass, part will be reflected and part refracted. The p-wave (TM wave) is reflected to a different extent than the s-wave (TE wave). The difference is dependent upon the angle ofincidence 1 (Figure 4.3). For manyangles ofincidence the reflection of the p-wave is somewhat suppressed relative to that of the s-wave. This causes the reflected light to be noticeably polarised. When this occurs, the refracted part of the incident light will also be partly polarised. Colour and the Optical Properties of Materials 132

incident p- (TM) wave reflected p- (TM) wave E kr Er B k Br

θ θ 1 3 n1

n 2 θ 2

Bt kt

refracted (transmitted) (a) p- (TM) wave

reflected s- (TE) wave incident s- (TE) wave kr E E r k B Br θ θ 1 2 n1

n2 θ 3

k Bt t refracted (transmitted) (b) s- (TE) wave Figure 4.2 Geometry of linearly polarised light with respect to the plane of incidence of the ray (the plane of the page): (a) p (TM) wave with E in the plane of incidence; (b) s (TE) wave with E perpendicular to the plane of incidence

With reference to Figure 4.3, the reflectivity or reflectance of the surface is given by Fresnel’s laws: 2 sinð12Þ Rs ¼ sinð1 þ 2Þ and: 2 tanð12Þ Rp ¼ tanð1 þ 2Þ for the s-wave and p-wave respectively. (Remember that 1 is equal to 3 for reflection.) These equations show that when light passing through air falls perpendicularly onto the surface of the material, with refractiveindexn, 133 Polarisation and Crystals

incident ray reflected ray containing s- and p- components θ θ 1 3 n1

n 2 θ 2

p- (TM) wave s- (TE) wave

refracted ray containing s- and p- components

u ¼ u Figure 4.3 The geometry of reflection that leads to the production of polarised light. Note that 1 3 and both the reflected and refracted rays generally contain s-wave and p-wave components. Polarisation perpendicular to the plane of incidence (the plane of the page) is represented by filled circles along the ray and polarisation normal to this direction as double-headed arrows. When both polarisation modes are present the symbols are superimposed

the angle of incidence, 1 is zero and the reflectivity of the p-wave equals that of the s-wave. The reflectivity of the s-wave remains almost the same as that of the p-wave up to an angle of incidence 1 of about 20 . Thereafter, the reflectivity of the s-wave smoothly increases to 100 % at grazing incidence (1 ¼ 90 ). The reflectivity of the p-wave diverges from that of the s-wave and decreases as the angle of incidence increases until, at a particular angle, it becomes zero (Figure 4.4). At this point the reflected beam is polarised to its maximum extent. This optimal angle of incidence is given by Brewster’s law:

100 s-wave p-wave

50 Reflectance / %

0 0 30 60 90 Angle of incidence / o Figure 4.4 Reﬂection at a glass surface (n ¼ 1.52) in air showing the p-wave and s-wave components. The reﬂectivity of the p-wave component is zero at the Brewster angle, 57 for a glass–air interface Colour and the Optical Properties of Materials 134

n2 tan1 ¼ tan3 ¼ n1 where the angles are given in Figure 4.3, n1 is the refractive index of the initial medium that the light ray traverses and n2 is the refractive index of the medium causing reflection. Forglass with a refractive index of 1.52 in air (so that n1 ¼ 1 and n2 ¼ 1.52), Brewster’s angle will be 56.7 (Figure 4.5). As the angle of incidence increases past Brewster’s angle the reflectivity of the surface for the p-wave will increase smoothly to 100 % at grazing incidence (1 ¼ 90 ). Thus, at both perpendicular and grazing incidence the s-wave and p-wave behave identically and all of the light is reflected. Unpolarised light shone on a sheet of high-quality optical glass arranged at the Brewster angle will reflect 100 % s-wave polarised light and transmit about 42 % s-wave and 58 % p-wave light. A stack of transparent glass plates aligned at the Brewster angle make up a Brewster window, which transmits almost 100 % p-wave polarised light and reflects 100 % s-wave polarised light. When total internal reflection is considered, the reflectivity of the s-wave and p-wave components of the light beam will be angle dependent in the same way. The reflectivity of the p-wave will become zero at the ‘internal’ Brewster angle of (90 56.7) for glass, i.e. 33.3. This has important consequences for the long-distance performance of optical fibres. Polarisation caused by reflection is present in many natural phenomena. For example, a primary rainbow is polarised in a direction perpendicular to the arc. The reason for this is because polarisation is introduced at the reflection inside the raindrop. The form of the curve is similar to that shown in Figure 4.4, but 100 % reflectance occurs at the critical angle for water, 49 rather than 90 (Figure 4.6). The Brewster angle B at which the reflectance of the p-wave falls to zero for reflection at an internal water air surface is given by:

completely polarised reflected light unpolarised incident light s-wave

≈57° ≈57° air glass 90°

s- + p-wave

partly polarised refracted light Figure 4.5 Unpolarisedlightonreﬂection froma glassplateattheBrewsterangle (57) will produce a completely linearlypolariseds-waveandapartiallypolarisedrefractedwave.Polarisationperpendiculartotheplaneofincidence (the planeofthepage)isrepresented byﬁlledcirclesalongtherayandpolarisationnormaltothisdirectionasdouble- headed arrows. When both polarisation modes are present the symbols are superimposed 135 Polarisation and Crystals

100

s-wave p-wave

Reflectance / % critical angle Brewster’s angle

0 0 30 60 90 Angle of incidence / o Figure 4.6 The internal reﬂection of light in a water drop (n ¼ 1.33) in air showing the p-wave and s-wave components. The reﬂectivity of the p-wave component is zero at the Brewster angle, 37 for a water–air interface

nair 1 tan1 ¼ tan3 ¼ tanB ¼ ¼ nwater 1:33 B ¼ 36:9

The angle of reflection in the drop is close to 38 for the important rays which suffer a minimum deviation of about 138 (see Table 2.3). Thus, the reflected light is almost 100 % polarised, with the s-wave being the major component present. Although the human eye is unable to detect polarisation direction, many animals have this ability. When discussing invisibility in animals, a point not made, but of considerable importance, is that numerous predators can detect polarisation differences. Thus, although a jellyfish, say, may appear invisible to humans, it may well be visible to a predator because of surface reflection, as this will generate considerable degrees of polarisation that will change as the jellyfish moves.

4.3 Polars

Polars are devices which transmit light vibrating (mainly) in a single plane. This plane is referred to as the vibration direction or the allowed direction of the polar. Light can thus be made into a linearly polarised wave by passing it through a polar. Polars can be made in a variety of ways. A stack of thin ﬁlms arranged at the Brewster angle to form a Brewster window, as described above, is one such method. Polarised light can also be produced using prisms of certain crystals, such as calcite, described below. As might be anticipated, a grid of conducting metallic wires can also polarise electromagnetic radiation. The component of the E vector parallel to the wire will be absorbed, as it can excite the free electrons in the wire readily, while the component of the E vector perpendicular to the wires passes largely unhindered. Naturally, the grid of wires must be closely spaced compared with the wavelength of the radiation in order to function efﬁciently, which makes these devices particularly useful in the infrared and microwave regions of the electromagnetic spectrum (20 mm1 cm). Colour and the Optical Properties of Materials 136

Many organic molecules are able to interact with light in a similar way to a wire. Organic molecules are essentially composed of chains of carbon atoms, each linked to its neighbours by strong chemical bonds (see Chapter 8). The molecular feature that allows useful polarising behaviour is a long and more or less linear molecular skeleton. The electric field of the incident light which is parallel to the long molecular axis is strongly absorbed, while the electric field that is perpendicular to the long axis is only weakly absorbed. This is because the electrons forming the chemical bonds can readily distort along the molecular axis but are much more constrained perpendicular to the axis. Naturally, some bonding patterns reinforce this trend, and these are the molecules that are of greatest importance as polars. For similar reasons, it would be anticipated that carbon nanotubes would also show strongly anisotropic behaviour with respect to polarisation of light. Should such molecules be arranged at random, then no overall polarisation will be recorded, but if they can be aligned then light will be emerge from the system strongly polarised perpendicular to the molecular axis. Inexpensive sheets of polarising material are made in this way, with all molecules aligned parallel to one another. The first of these to be widely available was Polaroid, invented and developed by Land in the years following 1927. A common molecule employed for polarising films is polyvinyl alcohol (PVA), the polymerized form of vinyl alcohol (CH2¼CHOH), first used in Polaroid sheets by Land in 1938. These long molecules are embedded in a sheet of an inert polymer, which is treated with iodine and then stretched. The stretching aligns the polymer molecules and the iodine enhances light absorption. Such materials are known as dichroic sheet polarisers (Section 4.8). These polarising sheets generally absorb almost 100 % of the light component with the E vector parallel to the polymer axis and transmit about 65 % of light with the E vector perpendicular to the polymer axis. While not perfect, large-area polarising sheets are inexpensive and widely available. Polarising films are put to practical use in ‘Polaroid’ sunglasses, where the film is arranged so as to endow it with a vertical vibration direction. Reflected light contains a considerable proportion of light polarised parallel to the reflecting surface, the s-wave component. The Polaroid sunglasses eliminate this horizontal component and so considerably reduce glare. Similar devices, polarisation filters, are used in photography to reduce the glare caused by reflection at water surfaces and clouds, and polarisation due to scattering (Chapter 5). Exactly the same mechanism operates in nematic liquid crystals (Section 4.13). If two polars are arranged in tandem so that light passes through both (Figure 4.7), then the first polar encountered by the light beam is called the polariser (that is, the object that introduces the polarisation) and the

polariser analyser

unpolarised polarised incident light transmitted light

I cos2θ I0 0 light polarised in light polarised in vertical plane plane at θ to vertical

Figure 4.7 Normal light of irradiance I0 transmitted by a polar (at left) will emerge linearly polarised parallel to the vibration direction of the polar, marked as a double-headed arrow. When two polars are arranged in sequence, the ﬁrst polar is called the polariser (at left) and the second the analyser (at right). In such a case a beam transmitted by both polariser and analyser will have a linear polarisation parallel to the vibration direction of the 2 analyser. The irradiance will be given by I0 cos u. If the vibration directions of the polariser and analyser are perpendicular to each other then no light will be transmitted by the pair 137 Polarisation and Crystals second the analyser (that is, the object that determines the resultant polarisation). The light irradiance transmitted by a pair of polars was ﬁrst investigated some 200 years ago by Malus. When the vibration directions of polariser and analyser are parallel the light transmitted will consist of a linearly polarised wave with irradiance equal to that of the incident radiation. If the analyser is now rotated with respect to the polariser the transmitted irradiance will diminish according to the law of Malus:

2 I ¼ I0 cos where I0 is the incident irradiance and is the angle between the vibration directions of the polariser and analyser. The emergent wave will be linearly polarised in a plane corresponding to the vibration direction of the analyser (Figure 4.7). No light will be transmitted when the vibration directions of the two polars are perpendicular to each other. In this orientation the polars are said to be crossed. Apart from polars that transmit or absorb linearly polarised light, polars made of materials that transmit or absorb circularly or elliptically polarised light are commonplace. The polarised-light-absorbing filters used in photography to enhance contrast by reducing glare or reflection are mostly circularly polarised films.

4.4 Crystal Symmetry and Refractive Index

Gases, most liquids and some solids, such as glasses, are isotropic with respect to their refractive index. That is to say, the refractive index is the same irrespective of the direction taken by the light beam. This is not generally true for crystalline materials, which are more often anisotropic. The optical behaviour is found to depend upon the symmetry of the crystal. Note that here it is the internal symmetry which is important, not the external shape, called the morphology or habit. Symmetry is defined in terms of symmetry operators, which apply reflections, rotations and so on to the atomic and molecular components making up the crystal. From among the various symmetry operators, the presence of a centre of symmetry is of considerable significance from the point of view of optical properties. A centre of symmetry, at (0, 0, 0) transforms any point (x, y, z)to(x, y, z). Both crystals and molecules which do not have a centre of symmetry are termed non-centrosymmetric. The unit cell of a crystal is the smallest convenient volume of crystal which displays the symmetry of the crystal and, if extended in three directions (like building up a cube or pyramid from bricks), will produce the macroscopic crystal. It is characterized by three axes, labelled a (of length a), b (of length b) and c (of length c), and the angles between them, a, b and g, where a lies between b and c, b lies between a and c and g between a and b. For historical reasons, the classification of external symmetry led to the derivation of six crystal families, which later was refined into seven crystal systems (Table 4.1).

Table 4.1 The crystal systems

Crystal family Crystal system Unit cell Example Isometric Cubic a ¼ b ¼ c, a ¼ b ¼ g ¼ 90 Rock salt, NaCl a ¼ b 6¼ c a ¼ b ¼ g ¼ Tetragonal Tetragonal , 90 Rutile, TiO2 a 6¼ b 6¼ c a ¼ b ¼ g ¼ Orthorhombic Orthorhombic , 90 Stibnite, Sb2S3 a 6¼ b 6¼ c a ¼ g ¼ b 6¼ Monoclinic Monoclinic , 90 , 90 Tungsten trioxide, WO3 a 6¼ b 6¼ c a 6¼ b 6¼ g 6¼ Anorthic Triclinic , 90 Copper sulfate, CuSO4 5H2O Hexagonal Hexagonal a ¼ b 6¼ c, a ¼ b ¼ 90, g ¼ 120 Zincite, ZnO a a ¼ b ¼ c a ¼ b ¼ g 6¼ Trigonal or rhombohedral , 90 or Calcite, CaCO3 a0 ¼ b0 6¼ c0 a ¼ b ¼ g ¼ , 90 , 120 Dolomite, CaMg(CO3)2 a Trigonal (rhombohedral) crystals are often described in terms of an alternative hexagonal unit cell given in the second line of this box. Colour and the Optical Properties of Materials 138

Cubic (isometric) crystals like common salt (halite or rock salt) are isotropic materials. They exhibitthe same refractive index in all directions and behave in the same way as a glass with respect to light. In all of the other classes crystals are anisotropic. Tetragonal, trigonal and hexagonal crystals have identical refractive indices along the a- and b-axes and a different refractive index along the c-axis. These crystals display two principal values of the refractive index, or two principal (refractive) indices. In orthorhombic, monoclinic and triclinic crystals there are three principal values of the refractive index, related to three mutually perpendicular axes. These axes may coincide with crystallographic axes for orthorhombic crystals but not for monoclinic or triclinic crystals, which are characterized by nonorthogonal axes. The numerical difference between the highest and lowest values of the principal refractive indices is called the birefringence of the crystal. The refractive index encountered by a beam of light entering a crystal in an arbitrary direction depends upon the polarisation of the light and lies between the values of the highest and lowest principal refractive indices. Because of the reciprocal relationship between the refractive index and the velocity of light in a material (Equation 2.3), the direction with the lowest refractive index is often called the fast direction or the fast axis, while the direction along the highest refractive index is the slow direction or slow axis. The relationship between crystal structure and refractive index is described in greater detail in Section 4.6. This variation of refractive index with crystal direction is unsurprising. The refractive index depends upon the density of atoms in a crystal (Section 2.4). In cubic crystals the atom density averages to be the same in all directions, while in crystals of lower symmetry some directions contain more atoms than others. For example, in the tetragonal rutile structure of TiO2, chains of TiO6 octahedra run along the c-axis. This structural feature results in the atoms in the crystal being much less densely packed along the a- and b-axes than along the c-axis chains. The refractive index along a and b is 2.609, while along c it is 2.900.

4.5 Double Refraction: Calcite as an Example

4.5.1 Double refraction

Although a variation in refractive index with direction may not be surprising, the way in which crystals with structures other than cubic interact with light is certainly so. This is well illustrated by the mineral calcite. Calcite is a mineral form of calcium carbonate (CaCO3). The unit cell is trigonal with a ¼ 0.641 nm and a ¼ 101.9, but it is sometimes more convenient to refer to a hexagonal unit cell in which a ¼ 0.499 nm and c ¼ 1.71 nm. The form of optical interest is called Iceland spar, and is a particularly clear form of the material. Iceland spar crystals are easily cleaved into rhombohedra. If such a rhombohedron is placed over a line or mark, double images will form when the crystal is in particular orientations (Figure 4.8). This can be demonstrated with greater precision by examination of a black spot on a sheet of paper through such a crystal. In general, two spots will be seen on looking from above through the crystal (Figure 4.9a). The spots also appear to be at different heights within the crystal itself. One spot will appear to be undeviated in position with respect to the spot on the paper. The undeviated spot is formed by light moving through the crystal as if it were glass, and the ray producing this effect is variously called the ordinary ray, the O-ray or o-ray. If the crystal is then rotated the ‘ordinary’ spot will remain in placewhile the other will rotate in a circle about the ﬁxedspot (Figure 4.9b d). The ray causing this behaviour is called the extraordinary ray, E-ray or e-ray. The crystal is displaying the fact that it has two indices of refraction and the feature is called double refraction. If a (linearly polarised) polar is placed over the crystal and rotated, at ﬁrst one dot disappears and then the other (Figure 4.9e and f ). If the crystal is picked up and tilted, then the separation of the two dots will change; and if it is possible to look down the diagonals of the rhombohedron, in one case only one dot will be seen, that formed by the o-ray, no matter how the crystal is rotated about this diagonal. This direction is called the optic 139 Polarisation and Crystals

Figure 4.8 Double refraction by an ordinary unpolished rhombohedron of Iceland spar. If the crystal is rotated, then the separation of the two pairs of lines visible through the crystal will alter and in some orientations only a single line will appear

axis. In a normal cleaved rhobohedron of calcite the optic axis lies along the body diagonal which passes through the ‘bluntest’ pair of corners. These occur at the two corners where the faces which meet all show obtuse angles. These results suggest that the beam of light entering a calcite crystal in a general direction is split into two refracted rays which are mutually polarised perpendicular to each other. The crystal thus resolves an incident beam of unpolarised light into two linearly polarised components with vibration directions perpendicular to each other; the s-wave and p-wave components. When the beam enters the crystal, each of the two linearly polarised components experiences its own refractive index and produces two refracted rays (Figure 4.10). The o-beam (the s-wave) is undeviated and consists of light linearly polarised with a horizontal vibration direction parallel to the base of the crystal rhomb. The e-beam (p-wave) is deviated and consists of light linearly polarised with the vertical vibration direction perpendicular to that in the o-beam. No light is absorbed (in a perfectly clear crystal) and half of the incident intensity is found in each of the beams. A prism of glass (or any isotropic substance) will produce a spectrum when a beam of white light falls onto one of the faces (Section 2.6). If the prism is made of a uniaxial material such as calcite, two spectra can form (Figure 4.11). In general, an incident beam of unpolarised light will be split into two, an ordinary and extraordinary beam, each of which will produce a spectrum due to dispersion. Unless the two rays are widely separated, the violet portion of the upper spectrum, due to the e-ray, will overlap the red portion of the lower spectrum, due to the o-ray. In this case the observed spectrum will appear as if a white band has appeared in the centre of an otherwise abnormally elongated spectrum. If the prism is cut so that the beam travels along the optic axis then only one normal spectrum will form. Colour and the Optical Properties of Materials 140

(a) projection of (b) c-axis (optic axis)

o o

e e

direction of rotation

o o e e

(e) (f)

Figure 4.9 Schematic representation of the appearance of double refraction by a crystal of calcite placed over a black spot on a sheet of paper. (a)–(d). As the crystal is rotated, one image (due to the o-ray) remains stationary and one (due to the e-ray) rotates. (e), (f). A sheet of polariser placed with its vibration direction, indicated as a double-headed arrow, perpendicular to the projection of the c-axis (the optic axis) of the calcite causes the e-ray to disappear, while the same polariser rotated by 90 causes the o-ray to disappear

The compound eyes of the now extinct ammonites were composed of calcite crystals, with each facet of the eye made from a single crystal. (There are several thousand facets making up each eye.) To allow for image formation without double diffraction effects, the optic axis of the calcite (the crystallographic c-axis) was aligned along the long axis of the facet.

4.5.2 Refractive index and crystal structure

More information on double refraction can be gained by investigating the refractive indices experienced by the rays. As Iceland spar is trigonal it will possess two principal refractive indices, related to the crystallographic 141 Polarisation and Crystals

observer

e-ray o-ray projection of c-axis (optic axis)

71° 109°

spot Figure 4.10 The passage of a monochromatic beam of light through a cleaved prism of calcite. Normal light falling perpendicularly upon the bottom face of the prism (as in Figure 4.9) is split into two components with different polarisation. The o-ray, with a vibration direction in a plane perpendicular to the c-axis, indicated by ﬁlled circles along the ray, is undeviated. The e-ray, with a vibration direction in a plane which includes the c-axis, indicated by double-headed arrows, is deviated by about 6. The top and bottom cleavage faces of the prism are (101) planes with respect to the hexagonal unit cell, and the c-axis (the optic axis) is a body diagonal of the cleavage rhombohedron axes. It is convenient to use hexagonal axes, in which case the hexagonal c-axis is the optic axis. When unpolarised light is transmitted along the c-axis (optic axis) only one spot is seen and both polarisation components experience the same refractive index, no ¼ 1.658. When the unpolarised light is transmitted perpendicular to the c-axis (optic axis) it is resolved into two beams, one with a vibration direction perpendicular to c, the o-ray, experiencing a refractive index n0, and one with a vibration direction parallel to c, the e-ray, experiencing a refractive index ne ¼ 1.486. The values of no and ne represent the principal refractive indices of the crystal.2 The birefringence (that is, the numerical difference between the principal indices for calcite) is given by no (1.658) minus ne (1.486), that is 0.172. The direction with the highest refractive index, parallel to the c-axis, is the slow direction. When unpolarised light is transmitted in anyother direction, the crystal will show two refractiveindices, each of which will apply to one of the polarisation components of the incident light. One of thesewill always be equal 0 to no but the other one will depend upon the direction of the light ray and is variable, written ne. When the light c 0 beam travels parallel to the -axis ne is equal to no and there is only one effective refractive index c 0 for the material. When the light beam travels perpendicular to the value of ne is equal to ne. To understand this difference it is necessary to turn to the crystal structure of calcite (Figure 4.12). The 2 2 þ 2 structure can be thought of in terms of planar (CO3) ions and Ca ions. The (CO3) ions are arranged in sheets perpendicular to the optic (c-) axis. When the light beam travels down the optic axis (the

2 There are a number of conventions in use: no is also written as o, no, No, O, nO, NO, No; ne is also written as e, ne, Ne, E, nE, NE, Ne. Colour and the Optical Properties of Materials 142

(a)

red extraordinary incident beam violet ray

red ordinary ray violet optic axis

(b)

red incident beam unpolarized violet

optic axis Figure 4.11 (a) A prism made of a doubly refracting material such as calcite will produce two spectra with normal white light when the optic axis is perpendicular to the beam. The e-ray is polarised parallel to the optic axis and the o-ray is polarised perpendicular to the optic axis. (b) When normal light is propagated along the optic axis only one spectrum forms, as the o- and e-rays are not separated

optic axis (c-axis)

2– (CO3)

2 Figure 4.12 The structure of calcite (schematic). The planar (CO3 ) ions are arranged in layers perpendicular to the crystallographic c-axis (the optic axis). The direction of the groups alternates from one layer to the next. The þ Ca2 ions are omitted for clarity 143 Polarisation and Crystals crystallographic c-axis) the vibration directions of the E vectors lie parallel to the planes containing the 2 (CO3) groups. Interaction with the planar groups is strong. The light beam is slowed; that is, the refractive index is high and corresponds to no. When the light beam travels in a direction perpendicular 2 to the optic axis, one E vector component vibrates parallel to the planes containing the (CO3) groups and so experiences the same refractive index as before: no. The other E vector component vibrates normal to 2 the planes containing the (CO3) groups, is less impeded and is not slowed so much; that is, the refractive index is low and corresponds to ne.

4.6 The Description of Double Refraction Effects

4.6.1 Uniaxial crystals

In the case of amorphous materials, such as air, water or glass, and cubic (isometric) crystals the refractive indices experienced by the horizontally and vertically polarised components of the light are identical. This means that they behave identically. The two beams cannot be separated and it is simplest to say that only one refracted beam is present. This is not true with all tetragonal, hexagonal and trigonal crystals, but unless the two refractive indices are quite different then the double refraction observed is too small to be noticed casually. All crystals in these systems will have one optic axis, the crystallographic c-axis, and they are described as uniaxial. A beam of light entering such a crystal splits into two beams. One polarisation component experiences a refractive index no 0 and the other a refractive index that has a magnitude ne lying between no and ne. A convenient way to visualize this interaction and to determine the refractive indices encountered by the horizontally and vertically polarised components of a light beam is by way of a construction called the optical indicatrix. This is an ellipsoid with the dimensions of the mutually perpendicular axes determined by the principal refractive indices of the crystal. The optical indicatrix for tetragonal, hexagonal and trigonal crystals is drawn with thevalue ne taken as parallel to the c-axis of the crystals and no as perpendicular to it (Figure 4.13). If ne is greater than no then the crystal is termed optically positive, and if ne is less than no then it is optically negative. (As the refractive index of a cubic crystal is the same in all directions, the optical indicatrix is a sphere.) The fast axis for uniaxial negative crystals is along the optic axis and perpendicular to it for uniaxial positive crystals. In order to determine the refractive indices experienced by the polarised components of a light ray, the beam is projected onto the indicatrix. The polarisation of the incident beam is resolved into two perpendicular components, normal to the beam direction, which form the major and minor semiaxes of the elliptical cross- section of the incident beam projected onto the indicatrix (Figure 4.14). Within this elliptical section the polarisation directions can be chosen to be parallel to the no axis of the indicatrix and perpendicular to this. Thus, a beam travelling down the optic axis has both vibration directions of the polarisation lying parallel to the no axes in the indicatrix, which indicates that both polarisation terms will see only a single refractive index, no, and so will not be separated. A beam travelling perpendicular to the optic axis will have the vibration directions resolved along one of the no axes, and along the ne axis. The incident beam will split into two, as described above, each component polarised perpendicular to the other. The vibration directions of a beam at an arbitrary angle to the indicatrix are resolved parallel to no and in a perpendicular direction to this. The refractive indices encountered by the two polarisation forms can be read from the lengths of the semi-major and semi-minor axes of the ellipse so formed. It is seen that, no matter what angle to the optic axis that the incident beam makes, 0 it generates an elliptical cross-section in which one semi-axis is always no. The other semi-axis is ne, which has 0 a value between no and ne. The relationship between the magnitude of ne and the angle that the ray makes with the optic axis is: Colour and the Optical Properties of Materials 144

(a) optic axis (c-axis)

no no

(b) optic axis (c-axis)

no no

Figure 4.13 The optical indicatrix for a uniaxial (tetragonal, hexagonal, trigonal) crystal: (a) uniaxial positive, ne >no ; (b) uniaxial negative, ne

2 2 1 ¼ cos þ sin ð 0 Þ2 n2 n2 ne o e

c 0 When the light beam travels parallel to the value of is zero and ne is equal to no and there is only one effective c 0 refractive index for the material. When the light beam travels perpendicular to the value of is 90 and ne is equal to ne.

4.6.2 Biaxial crystals

Similar effects to those just described will be seen with crystals belonging to the orthorhombic, monoclinic and triclinic systems. In these cases, crystals exhibit three principle refractive indices, na (which has the smallest value), nb and ng (which is the greatest value). The crystals have two optic axes and are referred to as biaxial. The horizontally and vertically polarised components of a beam of light entering such a crystal encounter different refractive indices, with magnitudes lying between the lowest, na, and the highest, ng. However, the refractive index encountered by both polarisation components of a light beam directed along either optic axis is nb. There is not usually an intuitive relationship between the optic axes and the crystallographic axes. 145 Polarisation and Crystals

optic axis (c-axis)

incident ray

′ ne

Figure 4.14 A beam of light incident on a uniaxial indicatrix experiences two refractive indices given by the major and minor semiaxes of the elliptical cross-section of the perpendicular to the beam direction. One of these is 0 always no and the other is ne. For a beam directed down the optic axis both refractive indices are no, while for a beam perpendicular to the optic axis one is no and the other ne

As with uniaxial crystals, a convenient way to visualize the interaction of light with crystals and to determine the refractive indices encountered by the horizontally and vertically polarised components of a light beam is by way of the construction if the optical indicatrix (Figure 4.15). This is an ellipsoid with the dimensions of the mutually perpendicular axes determined by the principle refractive indices of the crystal. In order to determine

nγ

nα

nβ

Figure 4.15 The general form of the optical indicatrix for biaxial (orthorhombic, monoclinic and triclinic) crystals Colour and the Optical Properties of Materials 146 the refractive indices experienced by the polarised components of a light ray, the indicatrix is sectioned perpendicular to the beam direction and the refractiveindices read from the lengths of the semi-major and semi- minor axes of the ellipse so formed. In the case of a biaxial crystal it is seen that the incident ray, in one orientation, will generate a section which is circular and with a semi-axis equal to nb (Figure 4.16). The direction of the light, at an angle V to the ng axis in this case, deﬁnes one optic axis. Clearly, there will be another optic axis at an equal angle V (or Vg if it is necessary to stress that the angle is with respect to the ng axis). The crystal is deﬁned as optically positive when the angle between the two optic axes, 2Vg, is less than

nγ incident ray along one optic axis

nβ

nα nβ

(a)

nγ optic axis optic axis

nβ 2Vγ

nα

nβ

(b)

Figure 4.16 The optical indicatrix of a biaxial crystal. (a) A ray of light incident upon a biaxial crystal can give rise to a circular cross-section with radius nb because nb lies between na and ng. This direction deﬁnes one optic axis, which is perpendicular to the cross-section. (b) The second optic axis makes an equal angle to the ng axis and is constructed in a similar fashion 147 Polarisation and Crystals

90 and optically negative when 2Vg is greater than 90 . As with uniaxial crystals, the maximum values of the refractive indices are called the principal indices of the crystal and the difference between the principal indices, ng na for a biaxial crystal, is called the birefringence of the crystal. To summarise, in all crystals of symmetry lower than cubic the refractive index depends upon the direction of vibration of the light ray. Any ray not passing down an optic axis is resolved into two rays linearly polarised in two mutually perpendicular directions.

4.7 Colour Produced by Polarisation and Birefringence

Birefringence, as such, does not normally result in colour production. A strongly birefringent crystal of Iceland spar is clear when viewed in ordinary daylight. However, this changes when polarised light is involved, and many beautiful colours can be seen in thin plates of anisotropic crystals when examined using polarised light. A good example is provided by a sheet of mica3 placed between two polars and viewed in transmission by holding the sandwich up to a white light. Suppose that the polars are crossed. As the mica sheet is rotated with respect to the polars, four positions, at 90, will be found at which the mica sheet becomes dark. These are the extinction positions. When the mica sheet is midway between these positions it will seem to be brightly coloured. The colour seen is very sensitive to the viewing angle, but if care is taken to look at the foil without any change of viewing angle then the colour will be seen to remain unchanged as the mica sheet is rotated. It is only the overall intensity which changes. The colour observed will also be found to depend upon the thickness of the mica sheet, although the overall pattern of variation of intensity will be the same as that just described as the mica foil is rotated. The colours produced by birefringent ﬁlms are explained in the following way. The beam leaving the polariser is linearly polarised. On entering the crystal this beam will be split into two, the ordinary and extraordinary rays. Because of the difference in refractive index experienced by the ordinary and extraordinary rays, each will move at a different velocity in the crystal. The result is a phase difference between the two rays called the relative retardation, which will be different for each wavelength. The optical path length of each beam is given by:

½ ¼ ordinary ray d o dno ½ ¼ 0 extraordinary ray d e dne where d is the (real) thickness of the plate. The relative path difference p between these rays is given by:

¼ j 0 j p d no ne where only the positive numerical difference between the refractive indices is important. The relative phase difference D between the ordinary and extraordinary rays is:

p p D ¼ 2 ¼ 2 j 0 j p d no ne l0 l0 where l0 is the vacuum wavelength of the light. The maximum phase difference is for rays travelling along or perpendicular to the optic axis, in which case:

3 Mica is the name applied to a group of structurally related minerals that are generally monoclinic and so biaxial in nature. Here, the exact species of mica is irrelevant. Colour and the Optical Properties of Materials 148

2p D ¼ djnonej l0

The retardation between the two rays means that the light is now elliptically polarised, not plane polarised. On traversing the analyser, the elliptical polarisation will be resolved along two mutually perpendicular directions: one parallel to the vibration direction of the analyser and one perpendicular to this. Some light will now be transmitted, the amount depending upon the wavelength. The resultant colour production is an interference effect. Although two light beams polarised perpendicular to one another do not interfere or form interference patterns, two beams with parallel polarisation can. On meeting the analyser, only the electric field components of the ordinary and extraordinary rays parallel to the allowed direction will pass. The phase difference will result in interference because the resultant electric field vectors are now parallel in each ray, which fulfils the interference condition. This causes the image to take on a colour because some of the wavelengths of the white light spectrum will interfere constructively and so be enhanced, while some wavelengths will interfere destructively and so be diminished. The colour perceived will be the sum of the effects over the visible spectrum. The colour observed as a function of retardation is given in Appendix A3.1. If the crossed polars are now rotated to be in the parallel position without changing the orientation of the mica, the complementary colour will be seen (Appendix A3.1). The colour will vary as a function of the thickness of the plate because the retardation is a function of the distance travelled by the two rays. It will also vary as the orientation of the beams change with respect to the optic axis for the same reason. This same effect can be exploited to reveal stress and strain4 in an isotropic material. When a material is stressed the density will change slightly. If the stress is directional then the density will vary in a pattern which mirrors this. Thus, an isotropic material under stress can contain optically anisotropic regions. In the case of molecular materials, including polymer films, the molecules can also become partly oriented parallel to each other during stretching, which enhances the effect. If the material is observed between crossed polars, coloured fringes will reveal the stressed areas. In effect, the stress encodes information on the linearly polarised incident beam which is decoded by the analyser. The effect is easily seen. Take a piece of plastic film and look at it between crossed polars. Generally, nothing of interest will be seen. If you now stretch the film (technically subject it to a uniaxial tensile stress) brightly coloured areas will appear (Figure 4.17). The birefringence so produced in the now anisotropic film is called stress birefringence. This feature is widely used in glass blowing to make sure that residual strain is not present. A glass workpiece is viewed between crossed polars and the strained regions are revealed. If necessary, the piece can then be annealed (reheated at a moderate temperature) to allow the glass to flow slightly and so relieve the strain present. Before the advent of high-speed computers the strain in complex engineering components could be analysed by building them of clear plastic and viewing the stress and strain fields present using crossed polars. Regions of the structure containing high levels of stress show coloured fringes, the spacing of which indicates the stress gradients present. This phenomenon is also well known to car drivers who wear Polaroid sunglasses. The windscreens of cars are stressed in a predetermined way so as to avoid catastrophic failure if hit by a flying stone or similar object. Light reflected from a hot road will be partly polarised, as explained above. The Polaroid sunglasses act as an analyser and coloured fringes delineating the strained areas are clearly visible over the windscreen.

4 The result of a stress (a force or load applied to a material) is a strain (a deformation). 149 Polarisation and Crystals

Figure 4.17 A thin piece of polymer film used to wrap food stretched and viewed between crossed polars. The bright colours in the normally transparent film reveal regions of high strain in the film

4.8 Dichroism and Pleochroism

If uniaxial or biaxial crystals are viewed by transmitted linearly polarised white light, many will be seen to change colour on rotation. Uniaxial crystals may display two colours (dichroism) and biaxial crystals three colours (trichroism). The term pleochroism (more than the usual number of colours, many coloured) is frequently used generically instead of either dichromism or trichromism. Note that dichroism (pleochroism) is due to the fact that the absorption of linearly polarised light is a function of the polarisation direction, whereas colour due to birefringence (double refraction) is due to the retardation introduced by the refractive indices encountered by linearly polarised light. The trigonal mineral tourmaline provides a good example of pleochroism. Tourmaline is the name of a group of hexagonal minerals of complex general formula. The best-known tourmalines are the gemstone elbaite, Na(Li,Al)3Al6Si6O18(BO3)3(OH,F)4 and the iron-rich form, schorl, NaFe3(Fe,Al)6Si6O18(BO3)3(OH,F)4.The best samples from the point of view of dichroism are prepared from schorl, which is a negative uniaxial mineral in which the value of no varies from approximately 1.66 to about 1.672, ne from 1.633 to 1.64 and birefringence from 0.027 to 0.032. Plates of schorl about 1 mm thick and containing the c-axis will transmit most of the incident light with a vibration direction parallel to the c-axis (the e-ray) and absorb most of the incident light with a vibration direction perpendicular to the c-axis (the o-ray). If the crystal is illuminated with polarised light and rotated through 90 it will become alternately dark and light (Figure 4.18). The use of linearly polarised light ensures that the beam of incident radiation is, in effect, solely made up of the o-ray component when the vibration direction is perpendicular to the c-axis or the e-ray component when the vibration direction is parallel to the Colour and the Optical Properties of Materials 150

(a) c-axis

unpolarised light o

(b) direction of vibration of incident light

(c)

direction of vibration of incident light

Figure 4.18 Dichroism in tourmaline. (a) A plate of the mineral tourmaline, cut so as to contain the crystallographic c-axis (the optic axis), transmits linearly polarised light differently depending upon the direction of vibration. The relative transmission factors are shown by the orthogonal pair of double-headed arrows. (b). An observer positioned above the crystal which is illuminated from below with linearly polarised light will see either a dark crystal (b) or a clear crystal (c), depending upon the orientation of the slab with respect to the direction of vibration of the light c-axis. The o-ray is strongly absorbed, leading to a dark-brown grey or even black appearance, while the e-ray is weakly absorbed, producing a light brown or grey colour. The origin of this dichroism lies in the presence of the transition metal ions Fe2 þ and Fe3 þ present and intervalence charge-transfer between these ions in which an electron is transferred from an Fe2 þ ion onto a neighbouring Fe3 þ ion (see Section 8.10). The linearly polarised light is strongly absorbed when the vibration direction, that is the electric field, coincides with the orientation of the ion pairs involved in the charge transfer. The electric field can then be seen to aid the electron transfer considerably. When the vibration direction is perpendicular to this orientation the efficiency of the electric field in aiding the electron transfer is minimal. The light is not absorbed and the crystal remains clear. Because the effect is viewed in transmitted light, rather low concentrations of the transition metal ions are required for best effects. Ruby (aluminium oxide (Al2O3) containing about 0.5 % chromium oxide (Cr2O3), approximate formula Cr0.005Al0.995O3) belongs to the hexagonal system and is dichroic. If viewed in linearly polarised white light with the plane of vibration parallel to the c-axis (the optic axis) the crystal appears orange red. When rotated by 90 so that the plane of vibration is perpendicular to the optic axis the colour seen is purple red. This mineral is 151 Polarisation and Crystals

negative uniaxial with no ¼ 1.768, ne ¼ 1.760 and birefringence 0.008; this latter value underlines the fact that the magnitude of the birefringence does not directly control the observation of pleochroism. (The colour of ruby is examined in more detail in Section 7.10.) The reason for pleochromism is that the absorption of light is dependent both upon its direction in the crystal with respect to the optic axes and its state of polarisation. Thus, all uniaxial and biaxial crystals show pleochroism, as in these materials the absorption spectrum of a crystal with light polarised along one crystal axis is different to the absorption spectrum when light is polarised along the other axes. The magnitude of the effect does not depend upon the refractive indices or the birefringence of the crystal, but very strongly upon crystal thickness. Because the effect is that of absorption, thin plates of crystal will usually show little change in colour on rotation even for strongly pleochroic materials, although there are exceptions. One group of strongly dichroic materials are sheet polars made from aligned arrays of organic molecules. These show strong absorption changes even in thin ﬁlms. However, because all visible wavelengths are absorbed, the change is from a light grey to dark grey/black and spectral colours are not seen. In fact, many polymer ﬁlms possess quite large degrees of birefringence because the manufacturing process tends to align the long molecules in one favoured direction. This can be readily shown by examination between polars.

4.9 Nonlinear Effects

4.9.1 Nonlinear crystals

Imagine a bright green beam of light emerging from a 1 cm cube of a perfectly transparent crystal, with no apparent electrical or other connections to it. This is certainly one of the most spectacular ways of colour production and always impresses on first sight. What is happening is that a beam of invisible laser radiation in the infrared is being converted by the crystal into a beam of green light. There are a number of ways that this can come about. In this chapter, only pure undoped materials are considered and the phenomenon is called frequency doubling or second harmonic generation (SHG). It is often described in terms of adding two identical photons offrequency n together, to produce a single photon with double the frequency, 2n; that is, half the initial wavelength. However, this description will be avoided so as to differentiate the process from up-conversion, another way of adding two photons together so as to produce a photon of doubled frequency (Section 9.9). In up-conversion an impurity ion acts so as to achieve the optical transformation.5 Frequency doubling utilizes only the pure matrix and impurity dopants are not involved. Frequency doubling is a nonlinear effect. There are two vital ingredients needed for the manifestation of nonlinear effects in crystals: a high electric field and a matrix of the correct symmetry. Consider the electric field initially. A time-varying electric field, such as that of a light wave, causes electronic polarisation in the crystal (Section 2.3). For electric fields of 2 1 normal intensity (in sunlight, E0 10 Vm ), the bulk polarisation of a material P is a linear function of the electric field E:

P ¼ e0wE ð4:1Þ where e0 is the permittivity of free space and w is the dielectric susceptibility of the material. (To relate the equation to quantities applicable at optical frequencies, note that the dielectric susceptibility is given by:

2 w ¼ er1 ¼ n 1

5 The language of the literature is often less than precise and frequency doubling is often called up conversion. Colour and the Optical Properties of Materials 152

cubic, P ∝ E3

parabolic, P ∝ E2

Polarisation P linear, P ∝ E

Electric field strength E

Figure 4.19 The variation of the polarisation with electric ﬁeld strength for linear, quadratic and cubic dependence. At low ﬁelds all approximate to linear behaviour

where er is the relative permittivity and n the refractive index of a transparent phase.) This is easily understood in a qualitative fashion. A transparent solid has strongly bound electrons. The electric field displaces these slightly and the resulting displacement (i.e. polarisation) is a linear function of the field strength. Although this serves perfectly well for ordinary light sources, it is only a first approximation. More exactly, the polarisation can be written as a series:

ð1Þ ð2Þ ð3Þ P ¼ e0w E þ e0w E E þ e0w E E E þ ð4:2aÞ where w(1) is the linear dielectric susceptibility, w(2) the second-order dielectric susceptibility, w(3) the third- order dielectric susceptibility and so on.6 The polarisation is no longer a simple linear function of the electric ﬁeld. To a ﬁrst approximation the vector complexities of Equation 4.2a can be ignored and it can be written in scalar form as:

ð1Þ ð2Þ 2 ð3Þ 3 P ¼ e0w Eþe0w E þ e0w E þ ð4:2bÞ where E is the magnitude of the electric field. The first term is the ‘linear’ term and is the only term of relevance in traditional optics. The succeeding ‘nonlinear’ terms are important when light from ordinary sources is replaced by laser light pulses in which the 8 1 electric field E0 can reach a value above 10 Vm . In such pulses, the electron cloud surrounding the atoms in the matrix is considerably distorted. If the electron cloud deformation can be approximated as varying as the square of the field, then the second term becomes appreciable. If the variation is best expressed in terms of a cubic variation, then the third term must be considered. At low field strengths, all give an approximately linear response (Figure 4.19).

6 Equation 3.2 is a vector equation that is most easily manipulated mathematically by tensor methods in which the terms w(1), w(2) and w(3) are arrays of coefﬁcients. For our purposes, the vector quantities P and E will be treated as scalars and the terms w(1), w(2) and w(3) as single valued numbers. 153 Polarisation and Crystals

Nonlinear colour production

ε χ(1) ε χ(2) 2 ε χ(3) 3 P = 0 E + 0 E + 0 E

1 Dispersion 1 SHG: ω, 2ω 1 THG ω, 3ω

ω ω 2 mixing of 1, 2, ω ω SFG: 1+ 2, ω ω DFG: 1- 2

ω → ω ω 3. OPO, OPA: 3 1+ 2

4. THG via SHG + SFG

SHG: Second harmonic generation

SFG: Sum frequency generation

DFG: Difference frequency generation

OPO: Optical parametric oscillator

OPA: Optical parametric amplifier

THG: Third harmonic generation Scheme 4.1 Colour production using nonlinear effects

Although the electric field strength is important in the observation of nonlinear properties of solids, the observed polarisation of a crystal is also strongly influenced by crystal symmetry. In a centrosymmetric unit cell (one that possesses a centre of symmetry), electronic polarisation in one part of the unit cell is equal and opposite to that in another part of the unit cell. In such materials only the odd-order w terms w(1) and w(3) have nonzero values. In non-centrosymmetric crystals (those lacking a centre of symmetry), the second-order term w(2) has a nonzero value and all terms are relevant. It is these latter types of material that are generally known as nonlinear optical materials. Nonlinear effects have many implications for optical properties and are being widely explored with a view to applications in numerous branches of photonics and optical engineering. Here, only the importance of nonlinear effects in colour production is considered. The major effects in this category are summarized in Scheme 4.1. The commonest nonlinear crystals for these uses are often labelled by acronyms rather than chemical formula: ADP (ammonium dihydrogen phosphate, (NH4)H2PO4), KDP (potassium dihydrogen phosphate, KH2PO4), BBO (beta barium borate, b-BaB2O5), LBO (lithium triborate, LiB3O5), AGS (silver gallium sulfide, AgGaS2), AGSe (silver gallium selenide, AgGaSe2). Less-common nonlinear crystals are usually called by their scientific names: lithium niobate (LiNbO3) and lithium iodate (LiIO3).

4.9.2 Second- and third-harmonic generation

The nonlinear terms in the polarisation equation allow photons to be added and subtracted in certain speciﬁc ways to generate light frequencies not available from existing sources. Commonly, nonlinearity is used to generate light of double the frequency, second-harmonic generation (SHG), of the input wave, but light of triple the frequency, third-harmonic generation (THG), has also been achieved using the same technique. Frequency doubling and tripling (SHG and THG) comes about in this way. The electric ﬁeld associated with a light beam Colour and the Optical Properties of Materials 154 is not steady, but varies sinusoidally. This variation can be expressed in terms of the angular frequency (see Appendix A1.1) as:

E¼E0 cosðotÞ where E is the magnitude of the electric ﬁeld vector, E0 is the amplitude of the electric ﬁeld vector, o is the angular frequency of the oscillation and t is the time. If this is substituted into a scalar form of Equation 4.2b the magnitude of the polarisation P is:

ð1Þ ð2Þ 2 ð3Þ 3 P ¼ e0w E0 cosot þ e0w ðE0 cosotÞ þ e0w ðE0 cosotÞ þ ð4:3Þ

The polarisation, thus, oscillates in a complex way, depending upon how many terms are of importance in Equation 4.3. This equation can be rewritten as:

P ¼ A þ Bcosot þ C cos2ot þ Dcos3ot þ ð4:4Þ

An oscillating charge gives rise to an electromagnetic wave, and so each of the terms in Equation 4.4 can then be thought of as the source of such a wave. The first term, surprisingly, implies that a static electric field will form in a nonlinear material, when illuminated with a suitably intense laser beam. The second term gives rise to a wave of the same frequency as the initial wave (that is, o) and is the normal interaction dealt with in traditional optics. The main contribution to the constant B is w(1). The third-term constant contains w(2) and gives rise to a wave of double that frequency, 2o, which corresponds to SHG (Figure 4.20a). If w(2) is zero, as in all centrosymmetric crystals, C is zero and no SHG wave can form. The forth-term constant contains w(3) and gives rise to a wave of tripled frequency, 3o, triple-harmonic generation (THG) (Figure 4.20b). (Note that, from a practical point of view, frequency tripling is usually carried out rather differently, as described below.) The irradiance of thesewaves depends upon the measured values of the dielectric susceptibilities. When light from a suitable laser passes through a crystal with appreciable second-order dielectric susceptibility, two beams may emerge, with frequencies of o and 2o. (In fact the production of a second harmonic from a crystal when illuminated by a laser is usually taken as a good test for the lack of a centre of symmetry.) One of the first nonlinear crystals to be utilized for this purpose was potassium dihydrogen phosphate (KDP), which can

(a) optic axis ω ω 2ω input wave non-linear crystal output waves

(b) optic axis ω ω 3ω

input wave non-linear crystal output waves Figure 4.20 The (schematic) generation of (a) frequency-doubled (SHG) and (b) frequency-tripled (THG) output using a laser input and suitable nonlinear crystals 155 Polarisation and Crystals convert red 694 nm output from a ruby laser into ultraviolet light of wavelength 347 nm. The crystals used in semiconductor lasers (Section 10.9), such as gallium aluminium arsenide (GaAlAs2), are nonlinear materials, and the laser output can consist of both o and 2o waves in certain circumstances. Crystals of BiB3O6 have been used to obtain pure frequency tripling, as in this phase the centrosymmetric term w(2) is zero.

4.9.3 Frequency mixing

More complicated processes can also take place when more than one beam irradiates a nonlinear crystal. If two beams characterised by angular frequencies o1 and o2 are used, not only 2o1,2o2 (the second- harmonic frequencies from each beam), and 3o1,3o2 (the third-harmonic frequencies from each beam), but also o1 þ o2 and o1 o2 (the sum and difference frequencies) can all be produced. The production of the sum and difference frequencies (Figure 4.21a) is known as frequency mixing sum frequency mixing (SFM) or sum frequency generation (SFG) and difference frequency mixing (DFM) or difference frequency generation (DFG). It is analogous to the formation of ‘beats’ easily heard when two sound waves mix.

(b) optic axis

2ω ω1 1 SHG

ω1 + ω2 SFG

ω1 − ω2 DFG ω2 2ω2 SHG

input waves non-linear crystal output waves filter

SHG: Second harmonic generation

SFG: Sum frequency generation

DFG: Difference frequency generation

(b) optic axis

ω mirror

input wave non-linear crystal output waves

3ω 2ω ω output waves input waves

Figure 4.21 (a) The (schematic) generation of sum and difference frequencies as well as doubled (SHG) and tripled (THG) output using a laser input and a suitable nonlinear crystal. (b) The (schematic) formation of THG using SHG followed by SFG Colour and the Optical Properties of Materials 156

Frequency mixing can be understood in the following way. Suppose the crystal is irradiated with two beams simultaneously so that:

E1 ¼E01 cosðo1tÞ

E2 ¼E02 cosðo2tÞ

The electric ﬁeld in the sample is then:

E1 þE2 ¼E01 cosðo1tÞþE02 cosðo2tÞ

Substituting this into Equation 4.3 will produce a series for P which will contain terms containing the sum and difference of the frequencies, cos(o1 þ o2)t and cos(o1 o2)t. These are the formal source of the two output waves, one offrequency (o1 þ o2) and one offrequency (o1 o2). SFG is widely used to obtain waves that are not readily produced by existing lasers. For example, infrared waves from a CO2 laser (l ¼ 10.6 mm) 3 þ and a Nd :YAG laser (l ¼ 1.06 mm) can be combined in an Ag3AsS3 crystal, which is opaque in the visible but transmits infrared, to give an output wave with l ¼ 0.96 mm. THG is often accomplished by the use of SHG in tandem with SFG (Figure 4.21b). The ﬁrst process produces an output frequency 2o1, and the addition of 2o1 plus the unconverted beam of frequency o1 using SFG gives an output wave offrequency 3o1. A typical application involves the generation of ultraviolet light from infrared laser output which is then used in LIDAR7 equipment for atmospheric surveying and the measurement of atmospheric properties such as ozone content. For these purposes, Nd:YAG (yttrium aluminium garnet) or Nd:YLF (yttrium lanthanum ﬂuoride) lasers produce an initial output in the infrared wavelength range (1300 925 nm) which is frequency doubled by an LBO (b-BaB2O3) crystal to a wavelength range of 750 463 nm and then tripled by a following LBO crystal to give an output of 433 310 nm.

4.9.4 Optical parametric ampliﬁers and oscillators

Optical parametric ampliﬁers and optical parametric oscillators are devices which use a nonlinear crystal to produce and amplify awave from a speciﬁc input. From the previous discussion it is seen that the introduction of a pair of waves offrequencies o1 and o2 into a suitable nonlinear crystal generates the sum frequencyoutput o3. That is:

o1 þ o2 ¼ o3

In this process, two input frequencies unite to give a single output frequency. There is no inherent reason why this should not operate ‘backwards’, so that a sufﬁciently powerful initial pump beam offrequency op produces output waves of frequencies os, which is the desired lower energy output wave (called the signal) and a redundant lower energy output oi (called the idler wave). On traversing the crystal, the powerful op input pump wave is gradually decomposed into two waves with angular frequencies os and oi as it crosses the crystal. Conservation of energy implies:

op ¼ os þ oi

The amount of light converted in a single pass is usually small, but is improved by repeated reﬂection within the nonlinear crystal in an oscillator design (Figure 4.22a). In an ampliﬁer, a signal wave of angular frequency os,

7 LIDAR is the optical equivalent of radar and is an acronym of light detection and ranging. 157 Polarisation and Crystals

(a) optic axis ω residual pump wave p ω ω pump wave p signal wave s ω idler wave i mirror mirror non-linear crystal

(b) optic axis ω ω residual pump wave p pump wave p ω ω amplified signal wave s signal wave s ω idler wave i non-linear crystal Figure 4.22 (a) The (schematic) operation of an optical parametric oscillator; an intense pump wave is converted into a signal wave and an idler wave. Repeated reflection increases the degree of conversion achieved. (b) The (schematic) operation of an optical parametric amplifier; input consists of a strong pump wave and a weak signal wave. Conversion of the pump into two waves, one of which matches the signal wave, results in amplification of the latter which has a low irradiance, is passed through a nonlinear crystal in conjunction with the powerful pump beam, op. The decomposition of the pump wave into an idler wave of frequency oi and signal wave of frequency os is identical to oscillator operation. The frequency of the newly generated signal wave, with a frequency os that matches the frequency of the input signal wave, adds to, and so amplifies, the signal wave (Figure 4.22b). In a suitable nonlinear crystal the refractive index encountered by the pump, signal and idler waves is a continuous function of the angle of the waves to the optic axis. This means that there are a range of allowed combinations of frequency generation that can occur. The actual frequencies generated, os and oi, will, therefore, vary as the angle of the incident beam on the nonlinear crystal is changed. This is known as tuning. The effect is considerable. For example, a commercial 3 þ oscillator using a b-BaB2O4 (BBO) crystal using frequency-doubled Nd :YAG laser output of 532 nm pump radiation can produce a signal wave varying in wavelength from approximately 650 to 1060 nm and an idler wave varying in wavelength from approximately 1060 to 3000 nm by rotation of the crystal over an angle of just 2.

4.10 Frequency Matching and Phase Matching

In principle, any non-centrosymmetric crystal can be used for the generation of other colours using harmonic and sum and difference methods. As the incident beams traverse the crystal they are gradually converted from one angular frequency to the others, such that:

o1 þ o2 ¼ o3 ð4:5Þ

This equation sets out what is known as the frequency matching condition, which must be fulﬁlled in all cases. However, the newly created waves are generally out of phase with each other and the incident beam. Beams with a phase difference will interfere with each other. A result of this interference, the intensity of the new rays emerging from the crystal is very low due to destructive interference. Destructive interference can only be prevented if all of the beams remain in phase. In the crystals that we are speaking of, this can be achieved by making the refractive indices and angular frequencies agree with the equation: Colour and the Optical Properties of Materials 158

n1o1 þ n2o2 ¼ n3o3 ð4:6Þ

This is known as the phase matching condition, which can be compared with the frequency matching condition given by Equation 4.5. As an illustration, consider SHG. In this case:

o1 ¼ o2 ¼ oo3 ¼ 2o

n1 ¼ n2 ¼ n

To satisfy Equation 4.6:

no þ no ¼ n32o which implies that the refractive index of the crystal for light of frequency o must match the refractive index of the crystal with respect to output with a frequency 2o. In an ordinary material, the refractive index decreases with increasing wavelength (Section 2.6). As:

p o ¼ 2 c l the refractive index at the frequency 2o will, therefore, be greater than that at o. Although obtaining phase matching would appear to be a tall order, it can be achieved in some uniaxial and biaxial crystals. Recall that if a beam of light is passed into a uniaxial or biaxial crystal it splits into two parts, the ordinary and extraordinary rays, each of which encounters its own unique refractive index (Sections 4.5 and 4.6). This provides a solution 0 to the problem. Take a hypothetical example. In a uniaxial positive crystal the refractive index ne encountered by the extraordinary ray is greater than the refractive index of the ordinary ray no. It is conceivable, therefore, that the refractive index encountered by the ordinary ray at wavelength l (angular frequency 2o) could be identical to the refractive index encountered by the extraordinary ray, with wavelength 2l at an angular frequency o (Figure 4.23a). It is then necessary to ﬁnd a crystal direction in which the refractive indices for the fundamental and the frequency-doubled beam match the phase matching angle. The same strategy can be 0 applied in the case of a uniaxial negative crystal, remembering that in this case ne is less than no (Figure 4.23b). Thus, the general principle is to use a uniaxial or biaxial crystal and to set the crystal at an angle to the incident wavesuch that the ordinary and extraordinary waves (that is, the o and 2o waves) encounter the same refractive index. Whether the input wave is taken as the ordinary or extraordinary component will depend upon the refractive indices of the frequency-doubling crystal. This strategy will not work with all crystals, but it is possible in some. The phase matching angle for second-harmonic-generated waves is given by:

uniaxial positive crystals; ne > no n ð2lÞ 2 n ðlÞ2n ð2lÞ2 sin2 ¼ e o o m ðlÞ 2 2 no neð2lÞ noð2lÞ

uniaxial negative crystals; ne < no n ðlÞ 2 n ðlÞ2n ð2lÞ2 sin2 ¼ e o o m ð lÞ 2 2 no 2 noðlÞ neðlÞ 159 Polarisation and Crystals

(a) no ne uniaxial positive ne > no Refractive index

λ 2λ Wavelength 2ω ω Angular frequency

(b) ne no uniaxial negative ne > no Refractive index

λ 2λ Wavelength 2ω ω Angular frequency

Figure 4.23 Phase matching. (a) Uniaxial positive crystal; the refractive index of the e-ray of wavelength 2l can be equal to the refractive index of the o-ray of wavelength l. (b) Uniaxial negative crystal; the refractive index of the e-ray of wavelength l can be equal to the refractive index of the o-ray of wavelength 2l

where m is the phase matching angle, no is the refractive index of the ordinary ray at wavelengths l and 2l and ne is the refractive index for the extraordinary ray at wavelengths l and 2l. (Specifically, these formulae are for ‘Type I’ phase matching, which is the condition that optimizes the birefringence of the nonlinear medium; see below.) Because the refractive index is temperature sensitive, crystals have to be placed in temperature-controlled cells to achieve reasonable amounts of conversion. This allows for an alternative method of tuning the output. Instead of rotating the crystal small amounts so as to achieve perfect phase matching, the temperature can be varied to obtain the same goal. Which of these alternatives is preferred will depend upon a number of factors. In some situations temperature variation is preferred to orientation variation. An important point has been glossed over in the description above the polarisation of the beams. The ordinary ray and the extraordinary ray passing through a low-symmetry crystal are polarised at right angles to each other and the polarisation direction is related to the direction of the optical axis (Sections 4.5 and 4.6). It is clear, therefore, that in all nonlinear optical devices (not just in SHG), not only must the beam directions with respect to the optic axis be precise, but the polarisation must also be correct. This leads to a number of different phase-matching schemes which quantify the relative polarisation of the pump, signal and idler waves with respect to the optical axis of the nonlinear crystal. (The designation of Type I phase matching above gives specific information on the relative polarisation directions of the input and output waves.) A second point of importance also needs to be mentioned. The paths of the ordinary and extraordinary rays in a nonlinear crystal are not parallel, but diverge (Figure 4.10). This effect is called walk off, and is quantified by the angle between the two rays, the walk-off angle. This serves as a means of separating the two rays, but also will drastically lower efficiency unless compensated. Colour and the Optical Properties of Materials 160

A ﬁnal point to note is that there are other ways of obtaining phase matching. Strictly speaking, the method described above is called birefringence phase matching. Clearly, it is limited both to birefringent crystals and to the subgroup of these that allow the refractive indices to be matched.

4.11 More on Second-Harmonic Generation

4.11.1 Polycrystalline solids and powders

The drawback of the method of SHG for the generation of colours so far described is that expensive large crystals are needed and precise phase-matching angles or temperatures must be maintained for intense second- wave output. Large crystals are needed because the output wave is generated successively as the input wave travels across the crystal. However, if a very high intensity output wave is not required then non- centrosymmetric power (polycrystalline) samples will generate second-harmonic radiation. The observed intensity will depend upon the crystal size. Small crystallites will not suffer from the destructive interference that necessitates phase matching. This makes materials that cannot be phase matched using birefringence available both for study and application. As the crystallite size increases, so the second- harmonic output will increase, but at some stage destructive interference will begin. This point is generally 8 speciﬁed by the coherence length Lc, which is the distance over which the o and 2o wavesbecome out ofphase by half a wavelength (p radians), given by: lo Lc ¼ 4ðn2onoÞ where l is the wavelength of the fundamental wave and n2o and no are the relevant refractive indices for the fundamental and frequency-doubled waves. Use of polycrystalline materials for SHG is widespread. Pure powders can be pressed into thin layers or fabricated as thin ﬁlms by a variety of techniques. Alternatively, particles can be mechanically distributed in glasses or other amorphous materials such as aerogels, or formed by the partial recrystallisation of glasses. This approach yields materials which have the formability of glass yet maintain SHG potential. A solid composed of many small grains (i.e. crystallites) will also give appreciable SHG output. This approach has been used with the important group of non-centrosymmetric III V and II VI semiconductors that includes GaAs and ZnSe. These arewidely used optoelectronic materials with high values of w(2), and although large crystals are available, they cannot be phase matched using birefringence. The use of a solid composed of partially oriented small grains gives rise to output waves that are all roughly (but not perfectly) in phase, resulting in a useful output. The generation of frequency-doubled visible light, say green light from the 1064 nm output of a Nd:YAG laser, makes optical microscopy possible. This technique has been used to image crystallites in glass and other amorphous materials. Naturally, SFG that leads to visible output can also be used.

4.11.2 Second-harmonic generation in glass

Glasses are centrosymmetric and the value of w(2) for any glass is zero. A glass, therefore, should not give rise to SHG. However, it is found that intense infrared laser light pulses with a wavelength of 1064 nm from a Nd3 þ :YAG laser sent down an ordinary commercial optical ﬁbre eventually produces SHG. After an hour or so a green light with a wavelength of 532 nm starts to appear along with the input infrared. As time goes by the intensity of the green light increases, and after 10 h or so is quite prominent.

8 This terminology is unfortunate. Traditionally, in optics the coherence length refers to the length of the wave train emitted by a light source in which the waves are all in step; in effect the length over which the wave can be considered to be a single sinusoid. 161 Polarisation and Crystals

How is it that a nonzero second-order dielectric susceptibility term has evolved during the infrared irradiation? The major sequence of steps occurring seems to be these. The intense electric field of the infrared radiation is strong enough to cause ions in the glass to migrate in a process akin to ionic conductivity. In the region near to the core cladding interface (see Section 2.9) the ionic displacements give rise to a permanent charge separation. The resulting permanent electric field Edc that builds up in the boundary region has been found to be as large as 108 Vm 1. It is this intense field that is partly responsible for the SHG. In addition, species migration results in the formation of defects in the glass. These contribute to the electric fieldandalsoresultinalossofsymmetry.Thetwoeffectsresultinanonzerow(2) termwhichinturnallows SHG to occur. Since the original observation, SHG in many glasses has been detected. For example, pure silica glass plates heated to a temperature of between 250 and 325 C and at the same time subjected to a static electric field of about 3 kV, which is maintained while the glass cools (called thermal poling), also show SHG in a region of about 3 mm close to the surface adjacent to the positive contact. This region has been found to be depleted in ionic constituents, and it seems that ionic movement in the applied voltage both establishes a permanent electric field at the surface and creates defects in the phase. Heating the glass at a temperature of a few hundred degrees in the absence of an applied voltage allows for ionic diffusion to re-occur, cancels the effect and returns the glass to its original nonactive state. Thermal poling is now frequently used to make SHG in homogeneous glass possible.

4.11.3 Second-harmonic and sum-frequency-generation by organic materials

Crystals that are built of organic molecules behave in exactly the same way as inorganic crystals and the considerations given above apply. However, organic molecules themselves can display nonlinear effects. The polarisability p of a molecule is written, in scalar form, as:

ð2Þ 2 ð3Þ 3 p ¼ p0 þ aEþa E þ a E þ where p0 is the permanent dipole (if any) on the molecule, E is the electric field, a is the molecular polarisability, a(2) (or b)isthefirst hyperpolarisability, a(3) (or g)isthesecond hyperpolarisability and so on. Clearly, a(1), a(2) and a(3) are the molecular equivalents of the macroscopic terms w(1), w(2) and w(3).Note that here E represents the field experienced by the molecule, the microscopic field. In general, this will differ from the external field applied to the collection of molecules, the macroscopic field, as it will include a contribution from the neighbouring polarised molecules. One great advantage of molecular nonlinearity is that the well-known techniques of organic synthesis can be used to modify the hyperpolarisability values at will. The addition or subtraction of polar groups, and their placement relative to the main body of the molecule, can all be adjusted precisely. In a crystal, the requirement for a non-centrosymmetric arrangement still applies for the case of SHG. However, molecules dispersed in liquids can show considerable bulk nonlinear effects if the molecules are partly aligned by an external electric field poling. The alignment need not be perfect to obtain appreciable bulk values of w(2). Similarly, nonlinear polymers or copolymers containing nonlinear molecules can be fabricated by alignment in an electric field during fabrication. As in the case of powders, below a certain thickness, phase matching becomes irrelevant for polymer films or molecular solutions. The overall polarisability P of the medium containing the nonlinear molecules is, to a first approximation, the sum of the contributions of the individual species present. For the second-order parameter w(2), for example, it is possible to write:

wð2Þ ¼ hað2Þiþ hað2Þiþ hað2Þiþ N1 1 N2 2 N3 3 Colour and the Optical Properties of Materials 162

3 að2Þ where N1, etc. are the number of molecules of type 1 (units m ) with molecular hyperpolarisability 1 and so on. The angle brackets represent the average value of the hyperpolarisability. In a crystal this average will simply be the value for a single molecule, but in a liquid the average will depend upon the temperature and the amount of thermal jostling each molecule undergoes. The equation can also apply to surface species (see below), in which case the units of N are m 2.

4.11.4 Second-harmonic generation at interfaces

Interfaces are non-centrosymmetric: one side of the interface differs considerably from the other. This means that atoms or molecules situated in the interface can be used for SHG because an atom or a molecule in the interface is exposed to quite different forces from one side compared with the other. This feature has long been exploited. Thus, at the surface of a metal the nearly free electrons act as the SHG oscillators, at semiconductor surfaces the incomplete bonds act in the same way, while many molecular species absorbed onto a surface can generate waves by virtue of the arrangement of bonds along the molecular length. Moreover, the SHG signals are fairly easy to detect because the bulk phases, be they solid, liquid or gas, generally are SHG inactive. Thus, it is routinely possible to detect less than a monolayer covering of a surface using SHG signals. There are many applications of the technique. Interfacial reactions, including absorption and desorption, corrosion, and the dynamics of electrode processes in electrochemical cells, are all open to study using interfacial SHG. The orientation of molecules absorbed onto a surface is also accessible with SHG. In interfaces, including biological cells, at which ordering takes place, strong signals can be generated, making optical microscopy possible. SHG allows the degree of chirality (i.e. right or left handedness; see Section 4.12) present to be determined.

4.11.5 Second-harmonic microscopy

The second-harmonic signal generated in a material can be used as the light source in optical microscopy, provided that the harmonic lies within the visible region, if the eye is the detector used. The technique has been mostly used in biologically oriented studies. Many molecules utilized in biological tissue are birefringent and are arranged into more or less ordered arrays, often at interfaces. These provide ideal environments for imaging. One advantage of using SHG is that light is not absorbed by the tissues, and so tissue damage that might occur with powerful illumination is avoided. Moreover, disordered or amorphous materials are not involved, making for greater discrimination in suitable subjects. Thus, SHG microscopy has been used to form high-resolution optical images of collagen and similar muscle tissue and the study of the retina in subjects suffering from the blindness-causing disease glaucoma (Figure 4.24). (See this chapter’s Further Reading, for more information.)

4.12 Optical Activity

4.12.1 The rotation of polarised light

One of the most intriguing results obtained by scientists trying to unravel the physics and chemistry of natural materials during the nineteenth century was the phenomenon of optical activity. For example, crystals of salts of the two acids tartaric acid and racemic acid were well known even hundreds of years ago and could be collected from old wine casks. The sodium salts of these two acids, sodium tartrate and sodium racemate, seemed to be chemically and physically identical. However, if linearly polarised light was passed through a solution of the tartaric acid salt the plane of polarisation rotated to the right as viewed by an observer looking towards the light source (Figure 4.25). The amount of rotation was as good a physical property of the compound 163 Polarisation and Crystals

Figure 4.24 Second-harmonic-generated image of an eye. The top left image is of a single ﬁeld of view from a single section. The top right image is a composite of overlapping ﬁelds to constitute an entire section. Several hundred sections were reconstructed using Almira software to produce the bottom image. The green light is SHG light from collagen and the red light is from two-photon emission from elastin-enriched collagen. [Reproduced with permission from Professor D. J. Brown, Eye Institute, School of Medicine, University of California, USA] Colour and the Optical Properties of Materials 164

light source polariser solution analyser observer Figure 4.25 Schematic diagram of the rotation of the plane of linearly polarised light by a solution of optically active molecules. The instrument used for the accurate measurement of the rotation is a polarimeter as, for instance, the melting point, and it could be used for characterisation purposes. The puzzle was that the corresponding salt of racemic acid was optically inactive and caused no rotation. The resolution of the problem was glimpsed when Pasteur made a painstaking optical examination of sodium racemate crystals. In 1848 he announced that these contained equal numbers of two forms, one ‘right- handed’ and one ‘left-handed’, meaning that they had the same relationship to each other as a left-hand glove to a right-hand glove or an object and its mirror image. Solutions of the two crystal types rotated the plane of polarisation by equal amounts, but in opposite directions. The compound sodium racemate could be described as a mixture of two forms of sodium tartrate, each of which rotated polarised light in equal and opposite directions. One of these was identical to the natural material described above, while the other appeared not to occur in isolation. The process of dissolution separates the solid crystals into molecules or ions. It was clear, therefore, that this example of optical activity was a feature which needed to be explained at a molecular level and not entirely at the crystallographic level. Since then it has long been established that any molecule that can exist in two mirror- image-related forms that cannot be superimposed one upon the other is optically active. They are referred to as chiral molecules. The mirror image molecules are called enantiomers and in organic chemistry are also known as optical isomers. Enantiomers, therefore, differ in one physical property: they display optical activity. One form of a chiral molecule will rotate the plane of polarised light in one direction and its enantiomer will rotate it in the opposite direction. The form of molecule which rotates the plane of polarisation to the right is labelled dextrorotatory. The form of molecule which rotates the plane of polarisation to the left is called laevorotatory. Enantiomers are efficient polarisation rotators. Mixtures of enantiomers in equal proportions will produce no resultant rotation of the plane of linearly polarised light and are called racemic mixtures, after the ‘racemic acid’ of Pasteur. In organic compounds, optical isomers occur whenever four different groups are attached to a tetrahedrally coordinated central carbon atom, making it a chiral carbon atom or chiral centre (Figure 4.26). Although it is not easy to see from drawings that the two structures cannot be superimposed, the construction of a simple model (putty and matchsticks) will convince you. The amount by which the linearly polarised light is rotated by an optically active material depends upon the number of chiral centres in the beam path, the wavelength of the light used to measure the effect and the t temperature. Optical activity is expressed in terms of specific rotation [a] l measured under standard conditions, which includes the wavelength l of the light used and the temperature t of the optically active material.9 For solutions, the specific rotation is given by:

9 Speciﬁc rotation is also termed the rotatory power r, but this is a poor descriptor, as the units are angle per unit length, not those of power, Watts. 165 Polarisation and Crystals

* chiral C atom NH2 NH2

CC **CH3 CH3 H H COOH COOH

m (b) (a) (S)-(+)-alanine occurs in nature Figure 4.26 The enantiomers of the amino acid alanine as examples of a chiral molecule. The chiral carbon atom in each molecule is marked C and is coordinated to the other groups by tetrahedrally arranged chemical bonds. Only one form occurs naturally, that in (a), (S)-( þ )-2-aminopropionic acid. The form in (b), (R)-( )-2- aminopropionic acid, can be synthesised

a ½at ¼ ð4:7Þ l dc where a is the rotation observed over a path length d (dm) for a solution of concentration c (g cm 3). For a pure liquid the concentration c is replaced by the density r (g cm 3) inEquation 4.7. For crystals, the specific rotation is measured as the rotation per millimetre of crystal. If the plane of polarised light is rotated clockwise when the observer looks towards the light source then the value of specific rotation is positive and when the rotation is anticlockwise the specific rotation is negative. Although a molecule with a single chiral centre exists as a left- or right-handed form, more complexity is introduced in molecules that contain several such centres. The results can then lead to increased optical activity (i.e. increased specific rotation), reduced optical activity or no optical activity at all. Tartaric acid, Pasteur’s crystals, is of this more complex type because the molecules contain two chiral carbon atoms. These can ‘cancel out’ internally in the molecule so that three molecular forms actually exist: the two optically active mirror- image structures, which cannot be superimposed on each other (laevorotatory and dextrorotatory), and the optically inactiveform, called meso-tartaric acid, which can be superimposed on its mirror image (Figure 4.27). The nomenclature is thus that if the optical activity is cancelled internally by the action of more than one chiral centre then the form is labelled meso-, as in meso-tartaric acid. If optical activity is lost because equal numbers of ( þ ) and () optically active enantiomers are present then the term used is racemic or racemate,asin racemic-tartaric acid. Inorganic molecules with tetrahedral or octahedral bond geometry can also form enantiomeric pairs. In addition, it should be noted that although many optically active crystals contain optically active molecules, this is not mandatory. In fact, the first instance of optical activity was noted in quartz by Arago in 1811. Crystals of quartz, one form of silicon dioxide (SiO2), occur in left- or right-handed forms although no molecules are present. Quartz is hexagonal (i.e. uniaxial), with the optic axis parallel to the crystallographic c-axis. In this material, the corner-shared [SiO4] tetrahedra that make up the crystal form helices (either right handed or left handed) along the optic axis. The plane of polarisation of linearly polarised light directed through a slice of crystal along the optic axis is rotated left or right, depending upon the handedness of the crystal. Similar light directed normal to the optic axis shows no change.

Colour and the Optical Properties of Materials 166

** *

laevorotatory mirror dextrorotatory inactive

COOH

OH H

chiral C * Figure 4.27 The three forms of tartaric acid. Two of these, the laevorotatory and dextrorotatory forms, are mirror images and are optically active. The third form is inactive. The chiral carbon atoms in each molecule are marked C. The chemical bonds formed by these atoms have a tetrahedral geometry

Enantiomers display identical physical and chemical properties except when they react with other chiral molecules. This has profound effects for life, because many biologically important molecules are chiral. Naturally occurring amino acids are ‘left handed’, while naturally occurring sugars are ‘right handed’. The molecules important to life on Earth are thus described as homochiral. (It seems that this bias is not restricted to life on Earth. Studies of the Murchison and Murray meteorites, reported from the late 1990s onwards, show that they also contain a preponderance of left-handed amino acids.) The differences in biological and pharmacological activity between two enantiomers can be pronounced. Drugs and pharmaceuticals derived from natural products are often chiral and the two enantiomers differ considerably in activity, one perhaps being beneﬁcial and one being nonactive or even toxic. The sensation of the odour of caraway, for example, is triggered by the left-handed enantiomer of limonene and that of mandarin oranges by the right-handed isomer. Vitamin C prevents the disease scurvy; the other enantiomer of this substance is biologically inactive. Such a list could be extended indeﬁnitely.

4.12.2 Circular birefringence and dichroism

The occurrence of optical activity is related to the polarisation of the incident wave. Taking the incident beam of linearly polarised light as made up of two equal and oppositely circularly polarised beams (Section 4.1), in a chiral material the electric ﬁeld is given by:

2i E ¼ ER þ e EL where R represents the right circularly polarised component and L the left circularly polarised component. The relative phase between the two polarisation modes is 2 and the orientation of the composite linear polarised 167 Polarisation and Crystals

beam is . In an optically active phase the refractive index of the left circularly polarised ray nL will be different than that of the right circularly polarised ray nR. The difference in refractive indices is given by:

Dn ¼ nL nR so that:

Dn is positive for nL > nR

Dn is negative for nL < nR

Thevalue of Dn, the circular birefringence, is characteristic of the material and is related to the speciﬁc rotation. The phase difference between the two polarisation modes, 2D, after travelling a distance d in the chiral medium is:

2pdDn 2D ¼ l0 where l0 is the vacuum wavelength of the light. The rotation that the linearly polarised beam suffers is D, given by:

p D D ¼ d n l0

D positive (nL > nR) represents dextrorotation (clockwise when the observer looks into the beam). D negative (nL < nR) represents laevorotation (anticlockwise when the observer looks into the beam). For a crystal, if d is measured in millimetres then the value of [D/d] is equal to the speciﬁc rotation t [a]l . There will be a slight difference in the absorption coefﬁcients of the incident right-hand and left-hand circularly polarised light as it travels through an optically active crystal:

Dk ¼ kL kR where kL is the absorption coefficient for left-handed polarised light and kR is the absorption coefficient for right-handed polarised light. By analogy with normal dichroism (Section 4.8), Dk is called circular dichroism. The specific rotation of a material is generally dependent upon wavelength. For many materials the specific rotation dispersion can be given by Boltzmann’s equation:

½a t ¼ A1 þ A2 l l2 l4 where A1 and A2 are experimentally determined constants. Optically active compounds are not coloured by virtue of this property. However, the optical activity is translated into colours when these compounds are viewed between polars. This is due to interference between Colour and the Optical Properties of Materials 168 the rotated and nonrotated components of the light beam after they have passed through the analyser, similar to that described with respect to thin crystal plates (Section 4.7).

4.13 Liquid Crystals

4.13.1 Liquid-crystal mesophases

Early experiments established that normal liquids were isotropic and had no effect upon the polarisation of light. Concurrent with this, it was observed that some organic crystals derived from cholesterol seemed to have two ‘melting points’. For example, cholesteryl benzoate appeared to show a lower ‘melting point’ (when the crystals turned into a cloudy liquid) and an upper ‘melting point’ (when the liquid became clear) separated by a temperature interval of 33 K. The cloudy region was described as consisting of one or more mesomorphic phases or, more usually now, mesophases. When studied with a polarising microscope the mesophase region, although certainly liquid-like, had a noticeable effect upon polarised light and seemed to behave rather like a low-symmetry crystal. Because of this, these curious materials were referred to as liquid crystals. The higher temperature clear liquid phase, formed above the second ‘melting point’, contains molecules which are truly random in direction and the liquid formed has no effect on polarised light. A century or more of investigation has shown that the liquid crystals first investigated are made up of needle- like or calamitic rigid molecules (from the Greek kalamos, reed). On raising the temperature a liquid crystal disorders in a number of steps, rather than all at once as is normal for a solid liquid transition. The structures which form within the mesophase region are complex and depend upon intermolecular forces and temperature, as well as on the geometry of the components. Above the first ‘melting point’ the molecules lose the strictly ordered intermolecular spacing typical of a normal crystal but still retain a partial degree of order. One of the commonest forms of disorder is found when the molecules retain a roughly parallel orientation but the geometric centres of the molecules are arranged at random as in a conventional liquid (Figure 4.28a). The preferred direction along which the molecules align is called the director. This structure defines a nematic liquid crystal, (from the Greek nematos, thread), so called because, when viewed in polarised light, long dark threads appear to occur throughout the bulk. These dark thread-like lines are optical effects caused by linear defects called disclinations which run through the structure. Most nematic liquid-crystal mesophases have only one optical axis and so are uniaxial. A considerable number of liquid-crystal phases retain the same roughly parallel orientation of the linear molecules, but they aggregate into sheets. This mesophase structure is that of a smectic liquid crystal (Figure 4.28b). This behaviour exactly parallels the situation to be found in Polaroid (Section 4.3). The effect on light is due to the presence long organic molecules, which act as uniaxial units and preferentially absorb one vibration direction of light comprising the incident beam. When the molecules are more or less aligned, as in Polaroid sheets or in liquid crystals, the effect of each molecule on an incident light beam is cumulative and the transmitted light is, to a greater or lesser extent, depending upon the degree of order, polarised. When the same molecules are randomly distributed the overall effect upon the polarisation of the incident light cancels and no polarisation is observed. More recently, liquid-crystal-like behaviour has been obtained from materials which are built up from disc- shaped molecules. To differentiate them from the calamitic molecules described above, disc-like molecular liquid crystals are called discotic. Although layers of molecules in a smectic arrangement are unknown to date in discotic mesophases, disordered columns of discs can occur to form columnar liquid crystals. Because nematic liquid crystals behave as uniaxial materials they can generate colours in polarised light in the same way as uniaxial crystals described above. However, some forms of liquid crystals, cholesteric or twisted nematic phases (described in Section 6.9), can produce colour directly by diffraction. 169 Polarisation and Crystals

(a)

director

(b)

director

Figure 4.28 The structure of calmitic liquid-crystal mesophases (schematic): (a) nematic; (b) smectic

4.13.2 Liquid-crystal displays

The orientation of the molecules in a mesophase is easily influenced by external disturbances such as electric fields. This has led to the most important use for liquid crystals in displays. These were first invented in the 1960s and subsequently developed intensively over the remainder of the twentieth century. Liquid-crystal displays (LCDs) were originally introduced on portable calculators and digital watches as black-on-grey images and are still widely used in clocks, watches, calculators and many other displays where the primary purpose is to display figures (Figure 4.29). Liquid crystals are not, themselves, coloured and do not emit light, so that external illumination is required. However, the molecules present in the liquid crystal have two important and exploitable properties. The molecules can alter the orientation of the direction of polarisation of the source light and the alignment of the molecules is easily changed by an externally applied electric field. This means that an electric field can easily change the orientation of an entry beam of polarised light. These attributes apply to a liquid-crystal film which is at the heart of these displays. The film is sandwiched between two glass sheets which are bounded by crossed polars (Figure 4.30a). White unpolarised background lighting provides the illumination. The liquid-crystal film is divided into pixels by grids of transparent conducting electrodes imprinted upon the glass sheets. When no voltage is applied to the electrodes the molecules in the liquid-crystal layer remain in the orientation originally imposed during manufacture. The polarised light beam from the entry polar is not rotated on passing through the liquid-crystal layer (Figure 4.30b). In this case the light is blocked by the exit polariser and the pixel appears dark. Applying a voltage to the electrodes causes the liquid-crystal molecules to rotate and in so doing to change the plane of Colour and the Optical Properties of Materials 170

Figure 4.29 Black-and-white LCD polarisation of the linearly polarised light beam by 90 (Figure 4.30c). The polarised light ‘follows’ the molecules and the plane of polarisation is rotated. It is then passed by the exit polariser and so the pixel looks bright. Use of colour filters extended this technology to include small-area colour displays on portable computers and digital cameras (Figure 4.31a and b). The relatively poor quality of these earlier screen displays (see figures) was less important than the fact that they were small, coloured and portable. Rapid improvements in performance has meant that at present, LCD screens are the norm for small portable displays such as cameras and phones, as well as computer displays and television and have replaced cathode-ray tube technology almost completely (Figure 4.31c). In colour displays each pixel is composed of three subpixels, each of which has a colour filter imposed before the final polariser. The fabrication of these colour filters is intricate and involves the dispersion of an organic pigment in a clear polymer substrate. In order to obtain a full-colour image three different pigments need to be used, corresponding to the three primary colours. Each of these is allocated to its appropriate subpixel. The colour seen by a viewer is additive and the primary colours used are red, green and blue. The display is usually viewed against a black background which enhances the colour contrast. There are two ways of controlling the output of the device. In a passive display, used for small screens and especially black-and-white displays on clocks and watches, the grid of electrodes is fixed and each pixel is ‘open’ or ‘shut’ depending upon the state of the voltage applied to the electrode grids. For colour displays, especially those showing moving images, such as television, the display is powered using an active matrix.Inthis technology, the fixed grid of ruled electrodes is replaced with a grid of transistors connected to each electrode. One transistor controls one pixel and each pixel can then be switched independently of any others. The light source in the display is ofimportance. As liquid crystals do not generate light, the light source can be daylight or artificial. When daylight is the source, as it is for many simple clock, watch and hand-held calculator displays, the liquid-crystal matrix is backed by a mirror. Daylight then traverses the unit and is reflected back again to the viewer. This simple solution is unsuitable for active displays such as television. In these, a light is provided behind the liquid-crystal matrix. The light then passes through the unit and is viewed directly. The quality of the light source is important in controlling the perceived performance of the screen. Direct-view 171 Polarisation and Crystals

(a) polariser on glass sheet outgoing light colour filter (optional) to viewer horizontal electrodes liquid crystal layer vertical electrodes polariser on glass sheet

incoming unpolarised white light

(b) liquid crystal molecules

vertical polariser

no light horizontally transmitted polarised light

(c)

light transmitted

Figure 4.30 LCDs (schematic). (a) The arrangement of the components in an LCD. (b) In one state, the liquid- crystal ﬁlm does not rotate the plane of polarisation of the light traversing it and no light is passed by the second polariser. (c) In a second state, the liquid-crystal ﬁlm does rotate the plane of polarisation of the light traversing it and light is passed by the second polariser Colour and the Optical Properties of Materials 172

Figure 4.31 (a, b) The colour LCD on a digital 35 mm camera (1999). The sharpness and colour rendition of the image is rather poor and pixels of the display are clearly visible. The LCD image size is approximately 3.5 cm 2.8 cm. (c) Liquid-crystal computer display (2004) showing far superior resolution and colour compared to (a) and (b). The display dimensions are approximately 34 cm 27 cm 173 Polarisation and Crystals

Figure 4.31 (Continued ) television screens have ﬂuorescent white light bulbs installed behind a light-diffusing screen, to give as uniform white light-emitting background. More recently, light emitting diode (LED) illumination (Section 10.8), behind the LCD, also with a light-diffusing screen, has been introduced. These give a wider colour gamut than ﬂuorescent back lighting, and also allow for a thinner screen.

Further Reading

Polarisation is described by E. Hecht, Optics, 4th edition, Addison-Wesley, San Francisco, 2002, Chapter 8. B. E. E. Saleh, M. C. Teich, Fundamentals of Photonics, John Wiley and Sons, Inc., New York, 1991, Chapter 6. A collection of classicpapers on polarised light, which includes reprints of studies by Huygens and Newton, and which makes fascinating reading is W. Swindell (ed.), Polarised Light, Dowden, Hutchison and Ross, Pennsylvania, 1975 (distributed by John Wiley and Sons). The easiest route to further information about polarisers is to visit the websites of optical component manufacturers. For example, enter ‘dichroic sheet polarisers’ into any search engine to ﬁnd up-to-date information. An extremely interesting (and well worth reading) account of the crystal structure of herapathite, the active polarising material in Polaroid, together with historical references, is B. Kahr, J. Freudenthal, S. Phillips, W. Kaminsky, Science 324, 1407 (2009). Crystal structures are described by J. V. Smith, Geometrical and Structural Crystallography, Wiley, New York, 1982. R. J. D. Tilley, Crystals and Crystal Structures, Wiley, Chichester, 2006. The relationship between crystal properties and light is summarized by F. D. Bloss, Crystallography and Crystal Chemistry, Holt Rinehart and Winston, New York,1971, Chapter 11. Colour and the Optical Properties of Materials 174

R. E. Stoiber, S. A. Morse, Crystal Identiﬁcation with the Polarising Microscope, Chapman and Hall, London, 1994. E. E. Wahlstrom, Optical Crystallography, 5th edition, John Wiley and Sons, Inc., New York, 1975. The information regarding ammonite eyes is given by R. Fortey, Life, Folio, London, 2008, p. 95; (originally published by HarperCollins, UK, 1997, and A Knopf, USA, 1998). A. R. Parker, In the Blink of an Eye, Free Press, London, 2003, pp. 217 220. The early studies of nonlinear optical properties of crystals are described in M. E. Lines, A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials, Clarendon Press, Oxford, 1977, Chapter 13. Reprinted in the series Oxford Classic Texts, 2001. Recent information on nonlinear optics is given in R. W. Boyd, Nonlinear Optics, 2nd edition, Academic Press, New York, 2003. Various authors, J. Mater. Chem. 19 (40), (2009); a single-topic issue on (mostly molecular) nonlinear materials. The observation of SHG in glass ﬁbres described in Section 4.11.2 is given in U. O¨ sterberg, W. Margulis, Opt. Lett. 11, 516 518 (1986). SHG in surfaces is reviewed by K. B. Eisenthal, Chem. Rev. 96, 1343 1360 (1996). Y. R. Shen, Nature 337, 519 525 (1989). The use of SHG in biological microscopy is surveyed by W. Mohler, A. C. Millard, P. Campagnola, Methods 29, 97 109 (2003). D. J. Brown, N. Morishige, A. Neekhra, D. S. Minckler, J. V. Jester, J. Biomed. Opt. 12, 024029 (2007). Optical activity is discussed in all textbooks concerned with organic chemistry and many concerned with inorganic chemistry. For example, see J. McMurry, Organic Chemistry, 6th edition, Thomson Brooks/Cole, Belmont, CA, 2004, Chapter 9. K. P. C. Vollhardt, N. E. Schore, Organic Chemistry, 3rd edition, W. H. Freeman, San Francisco, 1999. D. F. Shriver, P.W.Atkins, C. H. Langford, Inorganic Chemistry, 2nd edition, Oxford University Press, Oxford, 1994. An overview of chirality is given by G. H. Wagniere, On Chirality and the Universal Asymmetry, Wiley-VCH/VCA, Weinheim/Zurich, 2007. The homochiral nature of the molecules important to life on the Earth is discussed by A. Guijarro, M. Yus, The Origin of Chirality in the Molecules of Life, RSC Publishing, Cambridge, UK, 2008. Liquid crystals are described by P. J. Collins, Liquid Crystals: Nature’s Delicate Phase of Matter, Princeton University Press, Princeton, NJ, 1990. A simple description of active matrix LCDs will be found in S. Musa, Sci. Am. 277 (November), 87 (1997). More technical information is given in J. Hanna, I. Shimizu, Mater. Res. Soc. Bull. 23 (March), 35 38 (1996). There are a number of demonstrations of relevance to this chapter, including polarised light and nonlinear phenomena, available at http://demonstrations.wolfram.com/index.html. 5 Colour Due to Scattering

. Why is skylight polarised? . Why are eyes blue at birth? . How can yellow gold colour glass red?

Scattering, defined somewhat imprecisely, is the deviation of a beam of light from a straight path after interaction with an object. In this sense, refraction and reflection (and, in fact, many other optical phenomena) can correctly be regarded as scattering. In common parlance, however, scattering tends torefer to the interaction of light with small particles, often distributed at random in a continuous medium, such as small dust particles in air. Elastic scattering is a complex process that generally applies to the interaction of a light beam with specks such as smoke, dust or water droplets, in which little or no energy is exchanged. To a reasonable approximation, elastic scattering simply involves the redirection of light from its original trajectory into another one. It is elastic scattering which causes sunbeams to become visible in dusty or smoky rooms. Inelastic scattering describes the complex process occurring when there is significant energy exchange with the object, so that the scattered radiation has (generally) a lower energy than the incoming radiation. Intense colours can be produced by scattering. Some of the ways in which this comes about are described in this chapter.

5.1 Scattering and Extinction

As a beam of light passes through a transparent medium, a solid, liquid or gas, it gradually loses intensity, due to elastic scattering (Figure 5.1). The scattering particles might be the atoms or molecules that make up the medium, or else impurities of one sort or another within the medium. The gradual loss of intensity is generally

Colour and the Optical Properties of Materials Richard J. D. Tilley Ó 2011 John Wiley & Sons, Ltd Colour and the Optical Properties of Materials 176

Figure 5.1 A beam of light passing through a medium containing scattering centres will gradually diminish in intensity called extinction. Providedthat multiple scattering does not occur that is, each photon in the incident radiation is scattered only once as it crosses the medium the attenuation of a beam of light which has traversed a plate containing scattering centres follows the same exponential law given in Section 1.13):

I ¼ Io expðasxÞð5:1Þ where I is the irradiance leaving the plate, Io is the incident irradiance, x (m) is the thickness of the plate and 1 as (m ) is the (Napierian) linear scattering coefficient. The scattering length is defined as 1/aa. Equation (5.1) assumes that each scattered photon is lost from the forward-propagating beam. In point of fact, photons can be scattered in a forward direction as well as in any other direction. Forward scattering will tend to diminish the measured attenuation, thus reducing the apparent value of the absorption coefficient. The amount of beam attenuation obviously depends upon the number of scattering centres present. In addition, the total extinction that occurs is found to depend upon: 1. The ratio of the particle size to the wavelength of the light. Broadly speaking, large particles scatter more than small particles. 2. The ratio of the refractive indices of the particle and the surrounding medium. If the refractive index of the particle is the same as that of the medium, then no scattered radiation is registered, as has been mentioned previously (Sections 1.16 and 2.5). 3. The particle shape. Although calculations are difficult to make for geometries other than spheres, spheroids, rods and other shapes have been analysed. The degree of scattering depends upon the relative orientation of the particlewith respect to the illumination and the form of the particle, that is a long and needle-like rod will scatter differently than a short, thick rod.

5.2 Tyndall Blue and Rayleigh Scattering

In order to understand how scattering can lead to colour production the variation of scattering with wavelength must be investigated. The scattering of light by small particles was studied, from this point of view, by a number of scientists in the nineteenth century, but the most detailed experiments were made by Tyndall. He observed that liquids containing suspensions of small droplets, such as water containing a little milk, looked sky blue when illuminated with white light and viewed from the side. The beam of light responsible was also visible in the liquid and the light emerging in the beam direction took on a red hue (Figure 5.2a). The fact that a beam of light is visible in a suspension of small particles but invisible in a true solution is still the easiest way of distinguishing one from the other (Figure 5.2b). Tyndall supposed (correctly) that blue light was scattered more strongly than red light, and this blue scattering is still referred to as Tyndall blue. 177 Colour Due to Scattering

(a) bluish

white reddish

beam visible bluish

(b)

white white

beam invisible Figure 5.2 (a) Small particles suspended in a liquid will preferentially scatter blue light. The transmitted light will take on a reddish colour and the beam will be visible in the liquid. (b) In a true solution the amount of scattering is very small and the beam remains invisible

The ﬁrst important mathematical study ofscattering was carried out by Rayleigh, who investigated scattering by a small insulating (nonabsorbing) sphere with a diameter less than one-tenth of the wavelength of the incident light. Scattering by such bodies is referred to as Rayleigh scattering. The classical model for this scattering is that the incident electromagnetic wave (the light beam) causes the electrons associated with the scatterer to oscillate at the same frequency as the wave. The oscillating electrons then emit a wave with the same frequency (i.e. colour) as the incident wave, but in a different direction. In the case where a beam of unpolarised light of irradiance Io is scattered once only by a single scattering centre, the irradiance of the 1 scattered light Is is given by : p2 2 2 2 9 V m 1 2 Is ¼ Io ð1 þ cos Þð5:2Þ 2d2l4 m2 þ 2 where the measurement is taken at a distance d from the scattering centre, V is the volume of the scattering particle, l is the wavelength of the light incident upon the particle and m is the relative refractive index of the particle:

n m ¼ p nm

In this case np is the refractive index of the particle and nm the refractive index of the surrounding medium. For air nm is 1.0. The angle is the angle between the incident beam and the direction of the scattered beam

1 There are a surprising number of expressions for Rayleigh scattering to be found in textbooks and elsewhere. Many of these look quite different from one another. The expression in Equation 5.2 is the clearest for the present purposes. Colour and the Optical Properties of Materials 178

θ x

unpolarised incident beam

Figure 5.3 The Rayleigh scattering pattern of unpolarised light from small particles. The lengths of the arrows diverging from the small scattering centre can each be thought of as deﬁning the scattered irradiance at a distance d and at an angle u to the forward direction

(Figure 5.3). Note that these refractive indices only contain real parts n (the absorption index k is zero) and that the wavelength of the light impinging upon the particle should be that in the medium surrounding the sphere, rather than the vacuum wavelength. This is not important for air, but must be taken into account if scattering in glass or water is considered. The wavelength in a medium of refractive index nm is given by:

lvacuum lm ¼ nm

For example, if the scattering of yellow light of (vacuum) wavelength 550 nm from a small particle embedded in glass with refractive index 1.5 is considered, the correct wavelength to use in Equation 5.2 is 550/1.5 (that is, 367 nm), whilst if the same particle is suspended in water, refractive index 1.333, the correct wavelength to use is 550/1.333, or 412 nm. If there are Nv particles per unit volume, then the scattered irradiance from this volume is simply Is multiplied by Nv, as each photon is only scattered once. If the irradiance of the scattered light in a plane containing the incident beam, the scattering volume and the observer, the plane ofobservation, is plotted, then a characteristic Rayleigh scattering pattern or polar diagram is formed (Figure 5.3). It indicates that as much light is scattered backwards as forwards and that only half as much is scattered normal to the beam direction. As the 1/l4 term in the equation shows, all wavelengths scatter in this pattern, but the shorter wavelengths are more strongly scattered than the longer wavelengths. The importance of the equation went beyond simply explaining scattering. A comparison of measurements of scattering with the theory made it clear that molecules alone could operate as scattering centres. That is, even the purest gas would still show light scattering. Moreover, the formula allowed an estimate of molecular size and the number of molecules present in a unit volume of a gas to be made. These values permitted scientists to estimate Avogadro’s number and the molar masses of gases. Such information was of great interest towards the end of the nineteenth century, when the atomic theory of matter was still a topic of controversy.

5.3 Blue Skies, Red Sunsets

The blue colour of the sky has been a topic of interest since antiquity. Newton made the reasonable suggestion that it arose by reﬂection from small water droplets in the atmosphere. Rayleigh showed that it was due to 179 Colour Due to Scattering

1.2

1.0

0.8

0.6 visible

Relative scattered irradiance 0.4

0.2

300 400 500 600 700 800 Wavelength / nm Figure 5.4 Rayleigh scattering of visible light as a function of wavelength. Violet light is scattered approximately nine times more than red light scattering by gas molecules in the atmosphere. In the absence of an atmosphere the sky would appear black, even when the sun is high above the horizon, as indeed it does on the moon. The colour of the sky is the result of the wavelength differential inherent in Rayleigh scattering. Because this is proportional to 1/l4, violet light is scattered far more than red light (Figure 5.4). However, it is important to remember that all wavelengths are scattered in the Rayleigh pattern, as analysis with a prism will show (Section 2.6, Figure 2.10a). This suggests that when we look at the sky in a direction which is not towards the sun, the colour seen should be indigo or violet. In fact, the sky appears to be blue. This is for two reasons. First, the solar energy reaching the ground has less intensity in the violet than at longer wavelengths such as yellow and, second, the sensitivity of the eye to colour peaks in the yellow green region of the spectrum near to 555 nm (Figure 1.9). The result of these factors is that the sky away from the sun is perceived to be blue (Figure 5.5a). Towards sunset, when it is possible to look in the direction of the sun through a thicker layer of atmosphere, the scattering will remove blue light preferentially and the sun and sky will appear red (Figure 5.5b). The effect will be enhanced when this light is reﬂected at a shallow angle from clouds or ﬁne dust in the upper atmosphere, as can occur after volcanic eruptions, when spectacular sunsets are often recorded (Figure 5.6). This analysis suggests that the evocative phrase ‘blue remembered hills’2 requires further explanation. If the hills are far away, surely light scattered from them and entering the eye should be diminished in blue and hence look reddish. This ignores scattering of sunlight from the body of air that lies between the hills and the observer,

2 The earliest use of this phrase that I have located is in A Shropshire Lad, by A. E. Houseman, published 1896, poem XL ‘What are those blue remembered hills’. The hills are those in Shropshire, running along the Welsh Marches. The expression has also been used more recently by Rosemary Sutcliff, as the title of a memoir of childhood, and by Dennis Potter (in 1979) as the title of a play. Colour and the Optical Properties of Materials 180

yellow-white light from sun

scattered scattered light, seen as light, seen as blue blue

(a) observer in day time

light from sun, depleted in blue, seen as red

(b) observer at sunset Figure 5.5 (a) An observer during the day will see light from the sun as yellow white, and scattered light, in other directions, will give the sky a blue colour. (b) At sunset, light in the direction of the sun will be depleted in blue and appear red, while the sky overhead will remain blue due to the scattered light

which is blue enriched. This scattered light is called airlight. When hills are close, the amount of airlight is relatively small and the hills are normal in appearance. As the distance between the observer and the hills increases, the airlight scattering becomes dominant and the hills take on a blue indistinct appearance. The contribution of the airlight increases both as the sun rises in the sky and as the distance to the hills extends. Further hills look bluer. However, a stage will come when the hills eventually become obscured or invisible (Figure 5.7). A total eclipse of the moon was observed across much of Europe during the night of 21 January 2000. The moon was seen to be a copper colour. This distinctive colour is also due to light scattering in the atmosphere, as in the case of the red sky at sunset. A total eclipse of the moon occurs when the moon passes into the shadow (the umbra) of the Earth (Figure 5.8). The colour that the eclipsed disc takes on depends upon the light that is refracted by the Earth’s atmosphere to reach the surface of the moon. When the relevant part of the atmosphereisclearthiscanbeextensive.ThelightreachingthemoonisreddenedduetoRayleighscattering on its passage through the atmosphere. When reﬂected it gives the full moon a copper appearance. In less favourable conditions, when the atmosphere is cloudy, the amount of refracted light is low and the moon is barely visible when in the umbra. When the moon passes through the penumbra there is little reduction in light incidence and no colour changes are seen. Finally, note that a blue moon is also caused by scattering, but not by air molecules (see Section 5.6). 181 Colour Due to Scattering

Figure 5.6 Orange–red light reflected from clouds. The photograph was taken before sunrise on a clear winter morning. The sky is starting to take on a blue hue due to light scattered from the upper atmosphere. The clouds are lower in the atmosphere and reflect light that has travelled a significant distance through the air, and which has, as a consequence, become reddened

5.4 Scattering and Polarisation

A characteristic of Rayleigh scattering is that it produces strongly polarised light. Assume that an unpolarised light beam is travelling along the positive x-direction and that observations of the scattered light (at a distance d and angle to the positive x-axis) are made in the x y plane, the plane of observation (Figure 5.9a). The incident light beam can be resolved into two linearly polarised components: one with the electric ﬁeld vector lying parallel to the x y plane of observation and one with the electric ﬁeld vector lying

Figure 5.7 The foothills of the Pyrenees, France. The nearest ground is normal in colour, further hills appear blue due to scattered airlight and the furthest hills look indistinct and start to merge with the background Colour and the Optical Properties of Materials 182

penumbra

umbra

Moon Earth Sun

light refracted by Earth’s atmosphere Figure 5.8 A full eclipse of the moon is due to the moon passing through the shadow (umbra) of the Earth. The angles are greatly exaggerated here

(a) z

scattering centre

y d

θ x

incident beam unpolarised

(b) y total = perpendicular +parallel

electric field vector electric field vector parallel to plane perpendicular to plane of observation of observation

Figure 5.9 (a) A beam of unpolarised light travelling along the positive x-direction can be resolved into two linearly polarised components parallel and perpendicular to the x–y plane. This is taken as the plane of observation of the scattered light, which is recorded at a distance d and angle u to the positive x-direction. (b). The Rayleigh scattering pattern is made up of the sum of light scattered with its electric ﬁeld vector perpendicular and parallel to the plane of observation 183 Colour Due to Scattering

scattering centre completely polarised y partly polarised unpolarised unpolarised incident Ðx beam x

Ðy completely polarised partly polarised

Figure 5.10 The polarisation of scattered light is zero in the beam direction and at a maximum in the direction perpendicular to the incident beam. In other directions in the plane of observation it is partially polarised perpendicular to the plane of observation. The wave polarised perpendicular to the plane of observation is found to be scattered equally in all directions in the plane (Figure 5.9b). This wave contributes the term ‘1’ in the ð1 þ cos2Þ factor of Equation 5.2. The scattering from the component with the electric ﬁeld vector in the plane of observation has a dumbbell shape (Figure 5.9b). This wave contributes the cos2 term in the ð1 þ cos2Þ factor in Equation 5.2. The total scattering curve is the sum of both of these contributions. Thus, within the plane of observation, the scattered light is unpolarised in the beam direction, completely polarised perpendicular to the beam direction and partially polarised between these two directions (Figure 5.10). Sky light is polarised due to this differential scattering. The degree of polarisation is least (virtually zero) in the direction of the sun. However, sky light in a planewhich includes the observer and is at 90 to the line joining the observer to the sun is strongly polarised (Figure 5.11). Theoretically, the light should be completely polarised, but in reality it is found to be only about 75 85 % polarised. The reason for the discrepancy is that the actual polarisation observed at any point in the sky is a result of multiple scattering, the atmospheric conditions

light from sun

observer

horizon

light maximally polarised in this plane

Figure 5.11 Light scattered from small molecules in the air is optimally polarised in a plane at 90 to the direction of the incident radiation. It is not completely polarised in this plane due to multiple scattering. Observation of the polarisation of the light of the sky will allow the observer to estimate the position of the sun even on overcast days Colour and the Optical Properties of Materials 184 and the relative positions of the sun and the observer. In other directions the degree of polarisation lies between zero and this value. In reality, the accurate evaluation of the polarisation of the skylight is complex, and it is only in the second half of the twentieth century that accurate polarisation maps of the sky have been produced with the aid of computers. The polarisation of the sky can be observed using uniaxial or biaxial crystals. Cordierite (a magnesium aluminosilicate, Mg2Al4Si5O18, with the beryl structure) is a biaxial mineral and absorption of polarised light is strong along only one crystallographic axis. It is recounted that Vikings used this property of cordierite crystals, called sunstones, to locate the sun (and so navigate) even when the sun was not visible. The sky is viewed through a cordierite crystal which is rotated at the same time. If the direction of observation is in a plane perpendicular to the direction of the sun a clear patch of sky will appear alternately darkened and brightened as the crystal rotates. Viewing in the direction of the sun does not produce this effect, as the light is not very polarised. Hence, the direction of the sun can be determined even on cloudy days. The effect is easily checked with a piece of Polaroid ﬁlm. Humans are unable to detect polarised light,3 but bees and ants, and perhaps many other insects, possess this ability. They use this skill to navigate to and from the hive or nest even under conditions when the sun is hidden from them. The capability arises because the molecule responsible for photoreception in the eyes of all animals, rhodopsin (see Section 1.10), is a dipolar molecule with an optic axis. These molecules absorb polarised light energy maximally when the direction of polarisation is parallel to the optic axis of the molecule. In insects’ eyes these molecules are aligned in a ﬁxed direction, making them polarisation sensitive. In humans the molecules are free to rotate, so that the orientation of the optic axis is random and polarisation perception is lost.

5.5 Mie Scattering

Rayleigh himself extended scattering theory to particles of any size and shape, provided that the relative refractive index of the particle was small. Further work on this topic was carried out later by Debye and Gans, and the result is generally called Rayleigh Gans theory. This produces approximate expressions for scattering for a particle of arbitrary shape and size provided that the relative refractive index of the particle is small (usually just greater than unity) and the diameter of the particle is larger (but not too large) than the wavelength of the scattered light in the medium surrounding the particle. Although each element in the scattering particle is treated as a Rayleigh scatterer, the resulting angular distribution of scattered light differs considerably from that given by the simpler Rayleigh formula, Equation 5.2. Rayleigh Gans theory can be used, for example, to study the scattering of light by long polymeric molecules in solution. Despite this body of work, the most important advance was to apply electromagnetic theory to the scattering and absorption of light by an isotropic absorbing homogeneous sphere of any size. The mathematics of the problem is formidable and the ﬁrst complete theory was published in 1908 by G. Mie.4 Once again, though, each photon was presumed to be scattered only once. The theory is more general than that of Rayleigh, because it includes absorbing (i.e. metallic) bodies as well as insulators. It includes Rayleigh scattering as a special case for nonabsorbing small particles with a radius less than that of the incident light. Despite this universality, the term Mie scattering is sometimes reserved only for scattering by particles that are somewhat larger than those for which Rayleigh scattering is valid, say about one-third the wavelength of light or more.

3 This is not altogether correct. The visual phenomenon called Heidinger’s brushes, a faint small hourglass shape centred in the ﬁeld of view, is caused by the detection of polarised light in the retina of the eye. It is not observed by everyone. 4 In fact, considerable progress was previously made on the problem by Lorenz, and Debye published on this topic shortly after Mie, so that the theory is also called the Lorenz Mie theory, or the Mie Debye theory. 185 Colour Due to Scattering

Mie scattering theory allows for a complete solution to the scattering from a spherical particle provided that the optical constants of the material, n and k, in the complex refractive index:

N ¼ n þ ik are known (Section 2.1). Note that n and k are not constant but vary considerably with wavelength, making the computations arduous without a computer. It is found that for small particles the scattering is proportional to l p, with p ¼ 4, as in Rayleigh theory. As the particle size approaches that of the wavelength of light, p takes values between 4 and 0.2, while p ¼ 0 for the largest particles. In the Rayleigh scattering limit, the forward and backward lobes of scattered irradiance are equal (Figure 5.12a). As the radius of the particle approaches and passes that of the wavelength of light, the forward scattering lobe becomes dominant and the backward scattering lobe becomes negligible (Figure 5.12b). At larger particle sizes, forward scattering remains dominant, but side bands develop representing maxima and minima of scattering at deﬁnite angles (Figure 5.12c). The positions of these lobes depend upon the wavelength of the scattered light and so they can be strongly coloured. These coloured bands, referred to as higher-order Tyndall spectra, are dependent upon the particle size and so can be used for particle size determination. For the largest particle radii, wavelength dependence is lost. That is to say, large droplets scatter all wavelengths equally (although the scattering pattern still shows the strong forward-pointing lobe), which is the reason why fogs are white to the eye. These can be compared with the calculated scattering patterns for nonabsorbing spherical particles with a refractive index of 1.50, for a wavelength of light of 550 nm (Figure 5.13). With large particles, white light becomes reﬂected (rather than scattered as discussed here) evenly in all directions. This is the situation that holds in fogs and mists composed of fairly coarse droplets. Although Mie theory provides an exact solution for light scattering from spherical particles, the scattered irradiance pattern is a complex function of particle radius, the wavelength of the light and its polarisation, the refractive index of the particle and the refractive index of the surrounding medium. The resulting function describing the scattering is rarely written out in full, although approximate forms applicable to various special cases are to be found in the literature. Indeed, the scattering function is ‘neither simple nor intuitive’.

(a) incident beam

(b) incident beam

Figure 5.12 Light scattering by small particles (schematic). (a) Rayleigh scattering from particles much smaller than the wavelength of light. (b) Particles approaching the wavelength of light; the scattering becomes pronounced in the forward direction. (c) Particles larger than the wavelength of light; lobes appear which are wavelength dependent and so give rise to colours at speciﬁc viewing angles Colour and the Optical Properties of Materials 186

(a) (b) (c) 0.1 0.1 0.1 particle radius = 10 nm 50 nm 100 nm

(d) (e)

1.0 1.0 particle radius = 500 nm 1000 nm Figure 5.13 Mie scattering patterns for light of wavelength 550 nm from nonabsorbing particles of increasing radius with a refractive index of 1.5, in air. The scattered irradiance is proportional to the length of the arrows. Scale bars indicate the extent of scattering. Note that the scale in (d) and (e) is 10 that of the scale in (a)–(c)

On the other hand, the power of the Mie theory is that it allows the scattering to be evaluated via a calculation of the cross-sections for extinction Ce, scattering Cs and absorption Ca, where:

Ce ¼ Cs þ Ca

When absorption can be ignored: Ce ¼ Cs

These figures are often presented as efficiencies,orefficiency factors, Q: C C C Q ¼ e Q ¼ s Q ¼ a e G s G a G where G is the geometrical cross-sectional area of the particle projected onto the beam direction. For spherical particles, G ¼ pa2, where a is the sphere radius. The attenuation of a beam of light given by Equation 1.7:

Ix ¼ Io expðaexÞ can be expressed in terms of the extinction cross-section as:

Ix ¼ Io expðNvCexÞð5:3Þ and in the case where only scattering is important as:

Ix ¼ Io expðNvCsxÞ where Nv is the number of extinction (scattering, absorption) centres per unit volume. These equations can be written in terms efﬁciency factors, which, for spherical particles, are:

2 Ix ¼ Io expðNvpr QexÞ ð5:4Þ 2 Ix ¼ Io expðNvpr QsxÞ 187 Colour Due to Scattering

5 e

2 Extinction efficiency, Q 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Particle radius, / nm

Figure 5.14 Idealised sinusoidal curve for the extinction efﬁciency Qe for scattering of light of wavelength 550 nm by a spherical particle of TiO2 (rutile), with refractive index 2.755, in a transparent oil of refractive index 1.500

In general, the scattering cross-section of a particle Cs is proportional to the wavelength:

p Cs / l where p ¼ 4 in the limit for small particles that fall into the Rayleigh scattering regime and lies between 4 and 0.2 for particles near to or larger than thewavelength of the scattered radiation. A plot of log Cs versus log l has a slope of p and can be used to determine the scattering dependence. Ideally, a plot of the extinction efficiency of a particle Qe varies as a damped sinusoidal function (Figure 5.14). The principal and first maximum of the curve occur when the particle radius is about half that of the wavelength of the light being scattered. Unfortunately, the exact curves are not smooth, as portrayed, but have a pronounced ‘ripple structure’, which means that each wave of the sinusoid is made up of wavelets. In fact, these wavelets can be so severe as to obliterate the sinusoidal wave, and calculations are always needed to obtain accurate efficiency factors.

5.6 Blue Eyes, Blue Feathers and Blue Moons

When trying to explain almost all nonpigmentary biological colours, such as, for instance, blue eyes, it is difficult to tease apart the many optical effects that take place simultaneously. The tissues showing the colour are generally more or less transparent and contain thin films and reflecting surfaces. In the cases in which colour is believed to be primarily caused by scattering it is not a simple matter to describe the colour as due to either Rayleigh or Mie scattering. (In fact, the two terms merge into one another for particles of appropriate sizes and optical constants.) In such cases the term Tyndall scattering is a useful if rather imprecise expression that can used to describe the preferential scattering of shorter wavelengths of light by small particles or optical inhomogeneities in transparent materials that give rise to a visible blue colour. The hue so produced can then be called Tyndall blue. The colours generated in this way are rather ‘soft’, lacking in saturation, of rather low intensity and lacking the strong iridescence of multilayer colours (Sections 3.8 and 3.11). Colour and the Optical Properties of Materials 188

All eyes are blue at birth, and this is, in fact, a scattering effect. Of course, the whole of the eye is not blue, and the expression refers to the delicate colouring of the iris. This is essentially a transparent material consisting of a composite of various tissues, small crystalline regions and air vesicles, each of which have differing refractive indices. This inhomogeneity gives rise to preferential scattering of blue light. As the light transmitted through the lens and iris is absorbed in the underlying tissues, only the scattered light re-emerges, to give the impression of blue irises. After some weeks a pigment is laid down in many irises and it is this that changes the colour from blue to green or brown. The blue colours of many feathers from exotic birds are coloured blue in the same way. The outer part of the feather is a composite of several different more or less transparent proteins together with small air vesicles. The inhomogeneity causes preferential blue scattering. This is not easily seen against a background of reflected light, but if the feathers are backed by a dark absorbent layer, the blue colour becomes easily visible. As with feathers, blue scales on the wings of the male butterflies of the species Papilio zalmosis appear to be coloured by scattering which gives rise to Tyndall blue colour. The scattering is produced by a layer of air-filled tubes (alveoli) that penetrate a more or less transparent medium making up part of the scale structure and it appears that Tyndall blue coloration is the result. A similar phenomenon that falls neatly between nominal Rayleigh and Mie scattering domains results in the formation of the rare blue moon (or blue sun). This effect is caused by scattering in the atmosphere. Small particles, say of the order of 50 nm diameter, scatter violet wavelengths more strongly than red (Rayleigh). Large particles, say of the order of 10 000 nm diameter, scatter all wavelengths about equally (Mie). In between these limits some particle sizes, say about 7000 and 8000 nm diameter, scatter red wavelengths more strongly than violet. The light transmitted through a haze of these particles will then lose red light preferentially, giving the object, moon or sun, a blue green cast. The size and optical properties of the scattering particles must fall into a narrow and precisely defined range for the effect to be observed. This occurs rarely, but has been known to happen when forest fires inject uniform small oily droplets into the high atmosphere. At such times, a blue moon or sun may be spotted by a lucky observer.

5.7 Paints, Sunscreens and Related Matters

Many paints, plastics and glazes are made opaque by the addition of white pigment. Most frequently this is titanium dioxide (TiO2), but china clay and limestone are also commonly employed. These materials are not, in fact, white, but colourless. They give the appearance of whiteness when in powder form (in air) due to surface reﬂection and scattering. When mixed within a transparent matrix opacity is mainly the result of scattering. A similar opacity is found in opal glass, devitriﬁed glass, glass ceramics and porcelain, which contain varying amounts of precipitated crystalline phases in a glassy matrix. Both components are transparent in bulk form and the opacity comes principally from scattering by the inclusions. A control of the scattering power is essential if a satisfactory degree of opacity is to be obtained. The scattered irradiance Is, which determines the opacity, is given by Equation (5.4):

2 Is ¼ Io½1expðNvpr QsxÞ where Nv is the number of scattering particles per unit volume and x is the thickness of the pigment-containing layer. The idealized general form of the scattering efficiency curve plotted against size is a damped sinusoid (Figure 5.14). This shows that the maximum extinction efficiency is given by the first principal maximum of the curve. However, the calculation of scattering cross-sections or efficiencies is complicated, and, as the first principal maximum is of most importance, an approximation valid for nonabsorbing spheres, derived from 189 Colour Due to Scattering geometric optics (see van de Hulst in Further Reading), is useful. The equation takes the form:

4 4 Q ¼ Q ¼ 2 sin r þ ð1cos rÞð5:5Þ s e r r2 where r is given by: p ð Þ r ¼ 4 r m 1 l r is the sphere radius, l is thewavelength of the light in the medium surrounding the particle and m is the relative refractive index of the particle: n m ¼ particle nmedium

This equation holds when m is close to one (and is reasonable up to m ¼ 2) and when r is greater than l.The curve (Figure 5.15) is a useful approximation to curves obtained with exact computations, and is much simpler to handle. Note, though, that exact calculations can show any inherent ripple structure present; the simpler equation (Equation 5.5) does not reproduce this aspect of the scattering. The ﬁrst and highest maximum is at r 4.0, implying that the ratio of the particle radius r to that of the scattered wavelength l for maximum extinction is:

l r pðm1Þ

Substitution of typical values shows that extinction is at a maximum for a particle diameter approximately equal to the wavelength of the light scattered, but this depends upon the medium surrounding the particles. The majority of ordinary ceramic materials are produced by firing a mixture of finely grained powders or reactants that decompose to such ingredients. Much of the opacity of these bodies, which appear white to the eye unless pigments are deliberately added, is due to light reflection and scattering from the boundaries that remain between the crystallites of the final body. However, if the crystallites are of uniform size and are sintered so that they are in contact along the grain boundaries that separate one crystallite from another, the material regains transparency. This is the principle behind the fabrication of transparent polycrystalline ceramic bodies. One of the important successes of this approach was the fabrication of transparent polycrystalline alumina (Al2O3) tubes for sodium vapour lamps (Section 7.7). Sodium vapour is highly corrosive and reacts with silica e 3.0

2.5

2.0

1.5 Extinction efficiency, Q 10 20 30 ρ

Figure 5.15 Plot of Equation 5.5 for the extinction efficiency of spherical particles as a function of the parameter r. The first and most important maximum occurs at a value of r 4.0 Colour and the Optical Properties of Materials 190 glass, but does not attack alumina. Initial attempts to make transparent tubes were foiled because of the presence of small air-filled pores between each crystallite, which scattered light and rendered the ceramics opaque. The addition of magnesium oxide (MgO) as a sintering aid removed these pores and allowed transparent tubes, trade named Lucalox, to be manufactured and so allowed for the widespread introduction of high-pressure sodium street lighting. Scattering is also of immense importance in biological material, and transparency of biological materials has already been mentioned (Sections 1.16 and 2.5). The size of biological components spans the range from Rayleigh scattering (proteins, ribosomes, etc.) through traditional Mie scattering (bacteria, small cells) to reflection (large cells). The transparency or otherwise of much biological tissue depends upon the scattering of these components. Moreover, the appearance of skin and complexion, in both people and other animals, is closely related to surface and subsurface light scattering from structures lying within and on the surface of the skin, as cosmetics manufacturers are well aware. Sunscreens are a case in point. Titanium dioxide, as well as being a white pigment, is also a strong absorber of ultraviolet radiation (Section 10.1). It is widely used in sunscreen creams and lotions. However, it is considered undesirable that these products should be opaque. Fortunately, scattering drops towards zero as the value of r,or the ratio of particle radius to the wavelength of light, falls (Figure 5.15). Thus, the ultraviolet absorbing characteristics of titanium dioxide can be utilized without a high opacity if particle sizes are decreased. This is the region where Rayleigh scattering becomes dominant. Thus, small particles of titanium dioxide will not register any significant extinction and so not render the surrounding medium opaque at all. It is found that a particle diameter of about 20 nm is optimal. Particles of this size are invisible to the eye and provide best for the balance between transparency and absorption. This compares with maximum opacity for particles of titanium dioxide in paints and oils, which occurs when particles are approximately 200 nm diameter. 5.8 Multiple Scattering

Towhat extent can the examples in the previous section be treated in terms of Rayleigh or Mie scattering? These theories depend upon the assumption that each particle scatters only once, and this is surely not so in heavily loaded paints or sunscreens. Counterintuitively, the effects of multiple scattering can lead to increased transparency rather than increased extinction. For example, in experiments to make glasses containing lanthanoid5 ions for up-conversion purposes (Section 9.9), some slightly crystallized glasses were found to be more transparent than the original glass. The glasses were made from sodium oxide, aluminium oxide and silica (Na2O Al2O3 SiO2) together with small amounts of lanthanum triﬂuoride (LaF3). The initial glass is clear. Heat treatment causes crystals of LaF3 to form. Although some of these materials were opaque, due to the formation of large crystals in the matrix, others were more transparent than the parent glass according to the measured linear absorption coefﬁcient (Table 5.1). This was because the scattering centres were rather close together. Multiple scattering, particularly in a forward direction, from randomly distributed closely spaced scattering centres caused the diminution in the measured extinction. There is a second way in which multiple scattering can lead to greater transparency, most obviously displayed in the occurrence of transparency in inhomogeneous biological tissue. Perhaps of greatest importance (to us) is the evolution of the transparent cornea and lens of the mammalian eye. These structures are made of layers of different materials and would be expected to be opaque, just like the similarly constructed surrounding ‘white’ of the eye,the schlera. However, as the front part of the schlera is transparent, scattering has somehow been negated in this volume. This comes about by arranging the scattering material so that the scattered waves are out of phase with each other. The scattering is effectively suppressed and the material remains transparent.

5 The term lanthanoid now replaces the older term lanthanide (see Chapter 7 Footnote 7). 191 Colour Due to Scattering

a Table 5.1 The increase in transparency of glass ceramics containing LaF3 crystallites

Heating temperature/C Crystallite size/nm Absorption coefﬁcient/cm 1 0.075 750 7.2 0.060 775 12.4 0.070 800 19.6 0.095 825 33.3 0.101 850 opaque

a Data extracted from information given by M. J. Dejneka, Mater. Res. Soc. Bull. 23 (November), 57–62 (1998).

It should be noted that scattering centres can be arranged so that the scattered waves are in phase with each other and light intensity is reinforced. This may lead to strong colours. When scattering centres are arranged in such a way as to give rise to strongly reinforced or suppressed scattering they are found to be fairly regularly spaced. At this stage it is easiest to treat the problem by way of diffraction theory, which will be described later (Chapter 6).

5.9 Gold Sols and Ruby Glass

Gold sols, first deliberately prepared by Faraday utilizing the chemical reduction of gold chloride solution, are brightly coloured. The colour is due to microscopic crystallites of gold which are small enough to remain suspended in aqueous solutions. Ruby-coloured glass has been known for much longer, apparently dating to Roman times. It was produced regularly from the fifteenth century; the first published report on the production appears to be that of J. Kunckel, in 1689. The process involved gold as the ‘magic’ colorant. Slight modifications in the processing also allowed craftsmen to make blue or purple glass. Ruby glass is made by dissolving of the order of 0.01 % of gold in molten glass. If the glass is cooled in a normal way, which is fairly rapidly, then the glass remains clear, as isolated gold atoms are distributed evenly throughout the material. Colour is developed by annealing (reheating) the glass to 650 C for several hours. At this temperature the gold atoms in the glass aggregate to produce gold crystals with diameters of between 30 and 140 nm, distributed throughout the glass matrix. The colour is caused by these crystallites. Control of the crystallite size and, hence, colour by processing is difficult, which is why early glassworkers who had perfected recipes for the production of ruby glass guarded their knowledge jealously. A precise explanation of the colour of dilute dispersions of gold was given in 1908 within the framework of the theory derived by Mie for this purpose. In the previous discussions we have assumed that the particles which are scattering light are nonabsorbing insulators, allowing calculations to be performed using the ordinary refractive index n.6 However, metal particles are strongly absorbing, making the use of the complex refractive index:

N ¼ n þ ik mandatory (Section 2.1). For strongly absorbing collections of small particles it is found that, although the amount of light scattered is still proportional to V2l 4 (the Rayleigh dependence), the absorption of light is

6 Although this may be reasonable over very restricted wavelength scales for some compounds, it is not even valid for oxides such as titanium dioxide over a more extensive wavelength range. Colour and the Optical Properties of Materials 192

8 8

7 r = 25 nm 7 r = 50 nm

6 6

5 5

4 4 Qe 3 3 Qe Q Efficiency Factor, Q 2 Efficiency Factor, Q 2 a Qa 1 1 Qs Qs 400 500 600 700 400 500 600 700 Wavelength / nm Wavelength / nm

8 8

7 r = 75 nm 7 r = 100 nm

6 6

5 5

4 4 Qe Qe Q 3 3 s

Efficiency Factor, Q 2 Qa Efficiency Factor, Q 2

1 Qs 1 Qa

400 500 600 700 400 500 600 700 Wavelength / nm Wavelength / nm Figure 5.16 Mie calculations for spherical gold particles of radius r: (a) r ¼ 25 nm; (b) r ¼ 50 nm; (c) r ¼ 75 nm; (d) r ¼ 100 nm. The calculations were made using ‘Scatlab’ software (see this chapter’s Further Reading). [The optical constants for gold were taken from P. B. Johnson, R. W. Christy, Phys. Rev. B 6, 4370–4379 (1972)]

proportional to Vl 1, where V is the volume of the particles which are interacting with the incident light. Now, as V becomes smaller, the main interaction with light changes from scattering to absorption. Classical Mie calculations with spherical gold particles reveal how absorption and scattering changewith particle diameter. In the case of gold crystals with a diameter of less than approximately 50 nm, absorption is dominant and the total light removed by absorption plus scattering, the extinction, peaks at approximately 550 nm, in the green region of the spectrum (Figure 5.16a). The transmitted colour lacks this wavelength and imparts a ruby red colour to the glass.7 As the particle size increases, the scattered contribution becomes larger and eventually dominates the absorption (Figure 5.16b d). Owing to this shift, the perceived transmission colour also moves from ruby red towards purple and then blue. Further increase in size leads to the domination of scattering, and ultimately reﬂection, over absorption. Theglass loses bright colour and becomes opaque. Other colours can be produced in glass by using other noble metals, notably silver for yellow and platinum for pink.

7 The red colour of ruby gemstones and the synthetic ruby crystals used in the ﬁrst lasers arises for a different reason, and is treated in Chapter 7. 193 Colour Due to Scattering

Note that the calculations for gold are for spherical particles and use the refractive index data for bulk material. More refined calculations are now possible, for a variety of crystallite shapes (see this chapter’s Further Reading). These confirm the broad accuracy of the details given above. When the dimensions of the metallic particles fall below a diameter of 50 nm or so, absorption dominates the colour effects observed. Although Mie theory documents these changes, it does not explain them and is confined to spherical objects. The precise absorption characteristics of these small particles depend critically on the shape and are not well explained in terms of spheres. Further discussion of these colours is postponed to Section 10.12.

5.10 The Lycurgus Cup and Other Stained Glass

Ruby-coloured and similarly coloured glass was more or less reliably produced from medieval times using a variety of known recipes. The Lycurgus cup, an artefact that dates from the late Roman period, is unique and has not been duplicated. Like ruby glass, it is composed of glass coloured by metal nanoparticles but is renowned for its unusual colouring, as the glass is dichroic. In reﬂected light the colour is jade green, while in transmitted light it is a deep wine-red (Figure 5.17). The numerous small metal particles in the

Figure 5.17 The Lycurgus cup: (a) in reflected light; (b) in transmitted light. [Copyright Ó The Trustees of the British Museum] Colour and the Optical Properties of Materials 194 glassy matrix are polygonal with an approximate composition of 66.2 at.% silver, 31.2 at.% gold and 2.6 at.% copper. There are a wide range of particle sizes present, but most fall in the range 50 100 nm diameter, with an average of approximately 70 nm diameter. The colour is supposed to arise chiefly by the physical processes of scattering and absorption by the metallic particles embedded in the glass matrix making up the body. This suggestion can be evaluated to a first approximation by calculating the optical characteristics using Mie theory for spherical particles. The important parameters are the absorption efficiency Qa, the scattering efficiency Qs, the extinction efficiency Qe and the backscattering efficiency (also known as the radar backscattering efficiency) Qr. The extinction efficiency should correspond with the colour seen by transmitted light, while the backscattering efficiency should correspond to the colour seen in reflected light. Because the metal particles are an alloy of unusual composition, the optical constants were obtained by using the known optical constants of the pure metals silver, gold and copper, added in proportion to the alloy composition. Thus, the optical constants for the alloy, at wavelength l, are computed as:

nalloyðlÞ¼0:662nAg þ 0:312nAu þ 0:026nCu

kalloyðlÞ¼0:662kAg þ 0:312kAu þ 0:026kCu

The computed values of the efficiencies Qext, Qabs, Qsca and Qb (¼Qr /4p) for metal particles of 50 nm radius are a very close approximation to the experimental spectra taken from the cup (Figure 5.18). The results can thus be interpreted as supporting the idea that the colour of the Lycurgus cup can be explained in terms of Mie scattering from the alloy particles in the glass. It is noteworthy that neither pure silver nor pure gold particles of any size are able to reproduce the colours observed. The difference in colour between a typical ruby glass and the Lycurgus cup can be explained in terms of the relative amounts of scattering and absorption. The small particles which occur in gold sols, gold colloids and ruby glass are of the dimensions which exhibit high absorption and low scattering. The colour produced on transmission of daylight is white minus the colour absorbed by the particles, i.e. subtractive coloration. If the glass is examined in reflected white light it will look dark, as light is absorbed on entering the solid and little is returned to the eye. The surface may also be shiny, due to reflection from the glass matrix, and a slight ruby colour may be discerned due to light traversing the glass and being reflected back to the observer from the rear face of the object. The colour effects of stained glass windows in churches are similar to ruby glass. These glasses are coloured by subtractive coloration, often due to the incorporation of transition metal ions into the solid. Viewed in reflected light the glass appears dark and slightly shiny, for the same reasons as ruby glass. For this reason, stained glass seen from outside a church, for example, is always rather disappointing compared with the remarkable effects when illuminated by strong sunlight and viewed inside the building. As the particle size increases, scattering becomes more important and rapidly dominates the interaction of the metal particles with incident white light. In the Lycugus cup, scattering has reached this level. The colour scattered is mostly at the short (blue violet) end of the spectrum. The light traversing the glass will then appear depleted in blue and appear an orange red colour. In reflected light, little is absorbed, and the glass will be coloured by backscattered radiation from the metallic particles. This will be predominantly light from the blue end of the spectrum. The actual colours observed (in both ruby glass and the Lycurgus cup and related artefacts) depend strongly upon the particle composition, size and shape and on the density of particles in the glass. Thus, many subtle variations are to be expected in glass made by artisans using relatively irreproducible techniques. It would seem 195 Colour Due to Scattering

8 8 r = 50 nm 7 r = 25 nm 7

Qe 6 Qe 6

5 5 Qa 4 4 Qs 3 3

Efficiency Factor, Q 2 Qs Efficiency Factor, Q 2 Qa 1 1 Q b Q b

400 500 600 700 400 500 600 700 Wavelength / nm Wavelength / nm

77r = 75 nm r = 100 nm

Qs 55Qe 44Qe Q 33s

Efficiency Factor, Q 22Efficiency Factor, Q Qa Qa 11Q b Q b 400 500 600 700 400 500 600 700 Wavelength / nm Wavelength / nm

Figure 5.18 The extinction, scattering and absorption efﬁciencies (Qe,Qa,Qs) and the radar backscattering p efﬁciency/4 (Qb ), as a function of wavelength of the incident radiation, for particle radii in the range 25–200 nm that the craftsmen that fabricated the Lycurgus cup were both remarkably skilled and rather lucky on the day. For further discussion of these colours, refer to Section 10.16.

Further Reading

Classical scattering theory is treated in detail by H. C. van de Hulse, Light Scattering by Small Particles, John Wiley and Sons, Inc., New York, 1957 (reprinted by Dover, New York, 1981). C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles, John Wiley and Sons, Inc., New York, 1983 (reissued by Wiley-VHC, Weinheim, 2004). Atmospheric scattering, including the formation of blue moons, is in treated in C. F. Bohren, E. E. Clothiaux, Fundamentals of Atmospheric Radiation, Wiley-VCH, Weinheim, 2006. D. K. Lynch, W. Livingston, Color and Light in Nature, Cambridge University Press, Cambridge, 1995. Colour and the Optical Properties of Materials 196

Related references of interest with respect to atmospheric phenomena are J. Walker, Sci. Am. 260 (January), 84 87 (1989). J. Walker, Sci. Am. 238 (January), 132 138 (1978). The use of cordierite crystals for navigation, the navigation of insects using polarised light and information about the polarisation of sky light are given in A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientists and Engineers, Prentice-Hall, Englewood Cliffs, 1976, p. 369. J. Walker, Sci. Am. 238 (January), 132 138 (1978). R. Wehner, Sci. Am. 235 (July), 106 115 (1976). There are a considerable number of programs available to compute Mie scattering parameters. The original routine used, Mie Tab (which was available at http://www.zianet.com/damila), seems not to be compatible with newer operating systems. More recently I have used Scatlab, found at www.scatlab.com. Mie scattering calculations by Scott Prahl can be found at Oregon Medical Laser Center, omic.ogi.edu/calc/ mie calc.html. Others are available and can be quickly located via a Web browser. The Tyndall blue colour of Papilio zalmosis is described by J. Huxley, Proc. R. Soc. Lond. Ser. B 193, 441 453 (1976). The invention and production of Lucalox transparent alumina ceramics is described by J. E. Burke, Mater. Res. Soc. Bull. 21 (June), 61 68 (1996).

The discovery of enhanced transmission in LaF3 glass ceramics is reported by M. J. Dejneka, Mater. Res. Soc. Bull. 23 (November), 57 62 (1998). Opal glasses, polychromatic glass and other glassy materials are described from the inventor’s viewpoint by S. D. Stookey, Explorations in Glass, The American Ceramic Society, Westerville, OH, 2000. Scattering by biological tissues is described by S. Johnsen, A. A. Widder, J. Theor. Biol. 199, 181 198 (1999). The original paper of Mie on the colours of gold colloids is G. Mie, Ann. Phys. 25, 377 445 (1908). The microstructure of the Lycurgus cup is given by D. J. Barber, I. C. Freestone, Archaeometry 32, 33 45 (1990). 6 Colour Due to Diffraction

. What causes the colours reﬂected from compact discs (CDs) and digital versatile discs (DVDs)? . Why are opals coloured? . How do liquid-crystal thermometers work?

Diffraction is a particular form of light scattering. There is no hard and fast distinction between scattering and diffraction, although the term scattering tends to be used when discussing light interaction with small randomly distributed particles while the expression diffraction is associated morewith organized structures. In the case of Rayleigh and Mie scattering, the scattered waves have no implicit relationship to one another and the situation is called incoherent scattering. When the scattering object is made up of a more or less ordered arrangement of scattering centres, the scattered waves have a close relationship with each other, deﬁned, in part, by the separation of the scattering centres. Under these circumstances the outgoing waves can interfere constructively or destructively and the phenomenon is called diffraction. With this rather loose distinction preserved for convenience, one can note that the term diffraction also tends to be limited to the effects that occur when light waves interact with objects having more or less ordered features of a size similar to the wavelength of the radiation. After this interaction the waves travelling away from the diffracting feature, the diffracted waves, can interfere, thus giving rise to complex patterns of intensity. The result of diffraction is then a set of bright and dark fringes, due to constructive and destructive interference, called a diffraction pattern. When the separation between the scattering objects is less than the wavelength of light, similar effects still occur, but the classical diffraction equations are very restricted in application. The scattering is then often called coherent scattering rather than diffraction, although the two processes are identical.

Colour and the Optical Properties of Materials Richard J. D. Tilley 2011 John Wiley & Sons, Ltd Colour and the Optical Properties of Materials 198

Diffraction by a small pinhole was first described by Grimaldi in 1665. Since that time, many famous scientists have studied the diffraction effects arising when light passes through small apertures in an otherwise opaque screen. The mathematical analysis of the intensity patterns produced in this way was not trivial, but the solutions for apertures of various shapes, which agreed perfectly with observations, provided strong support for the wave theory of light. Classically, two regimes have been explored in most detail: (i) diffraction quite close to the object which interacts with the light, called Fresnel diffraction, and (ii) the effects of diffraction far from the object which interacts with the light, called Fraunhofer diffraction. An ordered collection of objects that diffract light, such as slits or circular apertures, atoms or molecules, etc. when arranged in a regular array, forms a diffraction grating and the mathematical analysis of diffraction gratings is an important constituent of the field of optics. Broadly speaking, the interference effects can occur after transmission by the collection, which then forms a transmission grating or after reflection from the collection, in which case it is a reflection grating. A transmission diffraction grating made up of a series of transparent apertures in an opaque materialproduces its effect by selectivelychanging the amplitude of the light passing through it and is known as an amplitude object and specifically an amplitude grating. Suppose, instead, that the grating is composed of adjacent strips of material which are transparent but of differing refractive indices. In this case a phase difference will be selectively introduced between the beams traversing adjacent regions, and the material is known as a phase object or phase grating. Differences in phase are not visible, but can be transformed into visible intensity differences using wave recombination techniques such as interference. Reflection gratings can also alter the amplitude or phase of the interfering beams to form amplitude and phase objects. It is diffraction that often sets a limit to the performance of optical instruments, including the eye, and the topic is, therefore, of particular practical importance. As well as allowing a quantitative analysis of the performance of optical instruments to be made, the mathematical analysis of diffraction led to the production of diffraction gratings for use in spectroscopy and the understanding of crystal structures, thus providing the foundations of much of modern science. More recently, diffraction studies have expanded into areas in which disorder, partial order or sub-wavelength order are dominant. These have important consequences in explaining the transparency of the cornea of the eye, for example The diffraction patterns formed by diffracting centres are sensitive to the wavelength of the incident light. When white light is involved, a multiplicity of such patterns form. When these are spatially separated, intense colours can be observed. Commonplace examples of this abound. Diffraction effects contribute to the shifting colours seen on many multicoloured wrapping papers and bags. The colours noticed in reflected light from the surface of a CD, and in some reflected patterns on banknotes or security logos, are also the results of diffraction.

6.1 Diffraction and Colour Production by a Slit

Tounderstand the diffraction pattern produced by a rectangular aperture (Section 6.2) it is easiest to beginwith a long, narrow slit. If such an aperture is illuminated by monochromatic light then some of the incoming wave is scattered by the edges of the slit and some passes through the central open part. The resulting waves interfere to produce a diffraction pattern on the side of the slit away from the source ofillumination. The pattern far from the slit, the Fraunhofer diffraction pattern, consists of a set of bright and dark fringes running parallel to the slit (if the complexities caused by the ends of the slit are ignored) (Figure 6.1). These are generally called orders. The first (straight through) bright fringe is called the zero-order fringe for constructive interference, because here the interfering waves have a phase difference of zero. Thereafter, the bright fringes are labelled as first order (phase difference l), second order (phase difference 2l) all for constructive interference. The same is true for the minima, which represent places where the waves that interfere are out of phase by a multiple of l/2. The first 199 Colour Due to Diffraction

diffraction angle θ 1st order

incident 0th order light

diffracted 1st order light

screen with narrow slit Figure 6.1 The fringes produced by diffraction of monochromatic light by a long narrow slit. Diffracted light is concentrated into bands at various values of the angle u to the undeviated beam minimum is called the ﬁrst-order fringe for destructive interference, with a phase difference of l/2 between the interfering waves, the second minimum is called the second-order fringe for destructive interference, with a phase difference of 3l/2, and so on. The irradiance pattern observed far from the slit (the Fraunhofer diffraction pattern) is given by the expression: sinx 2 I ¼ I ð6:1Þ x o x kwsin pwsin x ¼ ¼ 2 l where k is the propagation number of the wave (2p/l), w is the width of the slit, is the angular deviation from the ‘straight through’ position and l is thewavelength of the light (Figure 6.2). The positions of the minima (that is, the set of dark fringes) are given by:

x ¼p; 2p; 3p; ...¼mp ml sin ¼ min w

Ix /I0 1.0

0.8

0.6

0.4

0.2

θ/rad -1.5 -1.0 -0.5 0 0.5 1.0 1.5 l u ¼p Figure 6.2 The relative irradiance proﬁle (Ix/Io) for a single slit of width 3 , plotted over the range /2 Colour and the Optical Properties of Materials 200

screen

red 2nd order violet

red violet 1st order

white incident beam violet red 1st order

violet red 2nd order

Figure 6.3 The diffraction orders, resembling spectra, produced by diffraction of white light by a long narrow slit. The angle through which red light is deviated is greater than that by which violet light is deviated for each order of diffraction. These spectra are very weak compared with the central zeroth-order white fringe

where m takes values 1, 2, 3, etc. For min to be appreciable, the slit width w must be similar to the wavelength of the light l. Moreover, the formula shows that the spacing between the minima will be proportional to the reciprocal of the slit width, so that the narrower the opening the wider will the fringe spacing be. The width of the principal (central) maximum is twice that of the others. The positions of the maxima between the dark bands are not given by such a simple formula, but can be approximated by assuming that they lie midway between the minima. The sine of the angle through which a ray is diffracted is related to its wavelength. This indicates that each wavelength in white light will be diffracted through a slightly different angle, with red light diffracted through the greatest angle and violet light diffracted through the least. In this way, white light will produce a set of diffraction patterns, each belonging to a different wavelength (Figure 6.3). In the principal maximum, the spread of the red (long wavelength) waves will be greater than the spread of the violet waves. In the central part of the peak, all colours will overlap to give white. At the extreme edges, the red will extend further to give the fringe a reddish hue. The effect in the other maxima will be different as there is no overlap, as the different colours spread out. These patterns look like, and are called, spectra. They are referred to as first-order, second-order (and so on) spectra as they are recorded further and further from the undeviated beam. The intensities of these spectra are very low compared with that of the undeviated central fringe. They can be estimated by using Equation 6.1. Taking the central peak as irradiance 1.0, the first-order bright fringes have an irradiance of 0.0472. The other fringes are even weaker. (These spectra must not be confused with the intense spectra produced by diffraction gratings; Section 6.5.) Remarkably, the diffraction pattern of any object is identical to that of the complementary object that is, an object which is opaque, where the first object is clear. Thus, the diffraction pattern of a thin wire is identical to that of a thin slit. This is a statement of Babinet’s principle. It means, for example, that the central fringe in the diffraction patterns of both a thin wire and a thin slit are bright!

6.2 Diffraction and Colour Production by a Rectangular Aperture

A rectangular aperture is formed, conceptually, by shortening the length of the corresponding slit. The bright and dark ‘slit’ fringes will now form parallel to both the long and short edges of the rectangle. Thesewill overlap 201 Colour Due to Diffraction at the corners so that in some places bright fringes will coincide and in other places dark fringes will coincide, to reinforce the pattern. Elsewhere, bright and dark fringes will overlap to giveintermediate degrees of brightness. The irradiance is given by an equation almost identical to Equation (6.1): sinx 2 siny 2 I ¼ I xy o x y where all the terms have similar meanings to those in Equation 6.1, with y-coordinates substituted for x-coordinates where necessary. This produces diffraction maxima in the form of small rectangular spots running in two perpendicular directions (Figure 6.4a c). The intensity of the central rectangular spot is much greater than that of the others. The spot spacing is inversely proportional to the dimensions of the slit. Avertical narrow aperturewill giverise towidely spaced spots in a horizontal direction (parallel to the aperturewidth) and closely spaced spots in a vertical direction (parallel to the aperture length). If the rectangular aperture is replaced by a square, a square array of spots is formed. White light will produce coloured spots, as the diffraction angle is wavelength sensitive, as described above. Babinet’s principle allows one to state that the diffraction pattern of a rectangular speck is identical tothat of a rectangular hole.

(a)

(b)

(c)

Figure 6.4 Diffraction by a rectangular aperture: (a) aperture dimensions; (b) schematic diffraction pattern; (c) computed diffraction pattern for a slit of length 40l and width 30l. The spacing of the spots in (b) is inversely proportional to the aperture width w (horizontally) and height h (vertically) Colour and the Optical Properties of Materials 202

6.3 Diffraction and Colour Production by a Circular Aperture

The diffraction pattern of a circular aperture, a pinhole, is formed by the interference of light scattered from the periphery of the hole. The form of the diffraction pattern produced can be inferred by reference to a square aperture. In this latter case, a set of bright patches are formed parallel to the edges of the square. If the square is converted into an octagon, by cutting off the corners, it can be surmised that a set of bright patches will again form parallel to the straight edges. As the number of sides increases, so does the number of sets of bright spots. The diffraction pattern of a circular aperture takes this extrapolation to the limit. The pattern will consist of a series of bright and dark fringes concentric with the original aperture (Figure 6.5). The spacing of the maxima

(a)

(b)

(c)

Figure 6.5 The diffraction pattern from a circular aperture: (a) aperture dimensions; (b) schematic pattern, consisting of a bright central disc (Airy’s disc) surrounded by a set of circular light and dark rings (Airy’s rings); (c) computed image of the diffraction pattern. The radii of the rings are inversely proportional to the aperture diameter d 203 Colour Due to Diffraction and minima is given by:

ml sin ¼ d where is the angle between the directly transmitted ray and the diffraction ring, l is the wavelength of the light and d the diameter of the aperture. The computation of m requires rather sophisticated mathematics, first completed by Airy in 1835. The results show that m takes the values 0 (central bright spot), 1.220 (first dark ring), 1.635 (first bright ring), 2.333 (second dark ring), 2.679 (second bright ring) and 3.238 (third dark ring). As before, the intensity of the central spot will be considerably greater than the intensities of the surrounding rings. The angular separation from the centre of the pattern to the first dark ring is given by:

1:220l sin D ¼ ð6:2Þ d

The angular spread of the pattern increases as the pinhole gets smaller. The driving force for Airy’s work was the interpretation of the image of a star in a good telescope, Airy being Astronomer Royal at the time. Under ideal conditions the star will appear as a small point-like disc of light surrounded by diffraction rings. (These are much fainter than the disc and can often be more easily distinguished if the eyepiece is pulled in or out by a short distance so as to defocus the image slightly.) The central bright region is known as Airy’s disc and the surrounding circles as Airy’s rings. The performance of a telescope can be estimated by the appearance of these images. A distortion of Airy’s rings, when atmospheric conditions are good, is indicative of poor optics. Exactly the same effect will be seen if a point of light is observed in an optical microscope. Slight defocusing will reveal an expanding set of Airy’s rings, the perfection of which mirrors the perfection of the lenses.

6.4 The Diffraction Limit of Optical Instruments

When an object is imaged in an optical system, a telescope or a microscope, for example, it can be considered to be made up of innumerable point sources. Each of these will be imaged not as a point, but, in instruments with a circular limiting aperture, as a set of Airy discs. The idealized case is when the object consists of two small points, giving rise to two separated Airy discs. As the object points approach each other the image discs will begin to overlap. The resolution of the optical instrument can, therefore, be equated with the separation of the pair of adjacent Airy discs just before they appear to merge into one. There are a number of ways in which this can be translated into a number. Rayleigh suggested, for convenience’s sake, that this limit was taken as equivalent to the separation of point images when the Airy disc of one fell on the ﬁrst dark ring of the second. That is, the angular limit of resolution is:

: l D ¼ 1 220 d where D (radians) is the angular separation of the image pair, d is the diameter of the limiting aperture in the optical system and l is the wavelength of the imaging radiation.1 This equation (or similar) expresses the idea

1 This assumes that the optical components are perfect and ignores the resolution of the detector. In digital cameras the image resolution is generally limited by the resolution of the detector (quoted in megapixels) rather than by diffraction. Colour and the Optical Properties of Materials 204 that the ultimate quality of an image is limited by diffraction, the so-called diffraction limit of optical instruments. The form that this equation takes when applied to a particular instrument varies. For telescopes and binoculars, Equation 6.2 is taken as it stands. The value of d corresponds to the diameter of the objective lens or mirror and the value of D is usually quoted as an angle. Star catalogues, for example, list the separation of double stars in terms of their apparent angular separation. Clearly, the resolution limit of these instruments can be increased (nominally without limit) by increasing thevalue of the objectivelens or mirror diameter d. Hence, the drive to larger and larger telescopes, including, in recent years, the construction of telescope arrays with effective apertures of many kilometres in diameter. The great Mt Palomar telescope has a mirror of approximately 5 m diameter, giving the instrument an ideal diffraction-limited resolution of 2.8 10 2 seconds of arc. Cameras for photography of scenes far from the lens, and eyes, are similarly constrained. In these instruments, resolution is limited by the apparent diameter of the lens. For an eye, this equates to the diameter of the pupil. Taking a pupil diameter of 3 mm gives a diffraction-limited resolution of approximately 46 seconds of arc for light of 550 nm. Thus, the ideal eye cannot separate objects that have an angular separation less than this amount. Real eye performance is poorer than this. For example, the middle star of the ‘handle’ of Ursa major (the Big Dipper) star constellation, zeta Ursae majoris (Mizar), has a fainter companion, Alcor, at a separation of 11.8 minutes of arc, roughly 10 the diffraction limit of a perfect eye. This pair of stars, the ‘horse and rider’, is a ‘naked-eye double’, and the ability to separate them is considered a good test of eyesight (and of local atmospheric conditions). The optics of the microscope requires a different interpretation of Equation 6.2. In this instrument the diameter of the objective and the closeness of approach to the object are of importance. Moreover, the distance separation of the object points is of importance, rather than the angular separation. Keeping the same Rayleigh criterion that is, the Airy disc of one image point falls on the ﬁrst dark ring of the second leads to the equation: 1:220l s ¼ 2sini where s is the minimum separation of two self-luminous object points that can be resolved and i is the semi- angle subtended at the objective lens by one of the points. Note that the real situation in a microscope is more complicated, as the object points are rarely self-luminous. Abbe considered the problem in detail and produced the formula generally used for microscope resolution, which drops the factor 1.220 and takes into account the refractive index of the medium surrounding the points:

l s ¼ 2nsini where l is thewavelength of the light used, n is the refractiveindex of the medium surrounding the points and i is the semi-angle subtended at the objective lens by one of the points. The product ðnsiniÞ is the Abbe numerical aperture value of the objective, with values of 1.5 or better for good lenses. Ineither case, detail less than about half thewavelengthof the light used for observation is not recovered. This wavelength connection accounts for the drive towards the use of short-wavelength ultraviolet light for the preparation of integrated circuits on silicon chips to increase the density of packing of circuit elements. The recent development of ‘superlenses’ (Section 2.10) allows the diffraction limitation to be bypassed. Just as with the slit, the dependence of the diffraction angle upon wavelength means that a circular aperture illuminated with white light will produce a central white spot edged with red and a concentric set of coloured rings, rather like miniature circular rainbows. The formula indicates that each ring will have a violet inner edge and a red outer edge. 205 Colour Due to Diffraction

The same effect is seen when a beam of white light is scattered by a small mote of dust. Babinet’s principle indicates that this scattering will take the same form as that by a small pinhole. The central disc of the diffraction pattern will be bright in both cases and the colour sequence when illuminated with white light will be the same. The propagation of a beam of light is similarly affected by diffraction. A beam of laser light, for instance, will spread because of diffraction as it leaves the laser. This effect, although small, can be important, as when, for example, lasers beams are used to prepare optical masks or gratings (Section 6.5). Recently, in the same way that the diffraction limit for lenses has been bypassed, the diffraction spreading of laser beams has been nulliﬁed by patterning the emerging light to resemble the intensity proﬁle found in the Airy diffraction pattern from a circular aperture, emerging as an ‘Airy wavefront’ (see this chapter’s Further Reading).

6.5 Colour Production by Linear Diffraction Gratings

The simplest diffraction grating to visualize consists of a sheet of material inscribed with a set of regularly spaced parallel lines with a repeat period similar to that of the wavelength of light. Originally, gratings were made by carefully ruling lines on a metal sheet. This ‘master grating’ was replicated by making copies of the surface using a suitable dimensionally stable polymer film. These can act as reflection gratings when coated with a reflective metal such as aluminium. A mechanically simpler method of making a linear grating is to use the interference pattern formed by two laser beams. If these interact in a film of photoresist,2 then a sinusoidal interference pattern is formed which is transformed into a set of sinusoidal groves when the photoresist is processed. An advantage of this method is that the grating spacing is easy to control, because it is simply altered by changing the angle of the interfering laser beams and the wavelength of the laser light, both of which can be adjusted precisely. Light transmitted or reflected from such a linear diffraction grating forms a pattern of intense maxima separated by much weaker intensity oscillations. The positions of the maxima, called principal maxima, produced by a thin planar transmission or reflection grating in air are given by the grating equation:

dðsini þ sinmÞ¼ml ð6:3Þ where d is the grating spacing, i is the angle of incidence and m is the angle of diffraction of the mth-order line, taking values of 0, 1, 2, 3, etc. (Figure 6.6). (The convention when using this formula is that the angles of incidence and diffraction are considered to be positive when the incident and diffracted beams are on the same side of the normal to the grating and the angle of diffraction is negative when the diffracted beam lies on the opposite side to the grating normal to that of the incident beam.)3 Each value of m corresponds to a different diffraction maximum, called an order. For the zero-order diffracted beam, i ¼m, which is identical to straight through transmission or ordinary (specular) reﬂection. It is important to be aware that these orders are not the same as the weak orders from a single scattering object (as depicted in Figure 6.3, for example) but are new, intense peaks formed by the periodic grating. The most intense orders (after m ¼ 0, which is the most intense), are the m ¼1, then m ¼2. For a particular value of the orders form a row of diffracted maxima with spacing proportional to 1/d. Because each line on the grating acts as a contributing slit they are of much

2 A photoresist is a polymeric material that is altered by exposure to radiation, (usually light). After illumination, the photoresist is weakened or strengthened in those areas which were exposed to light. The weakened areas can be selectively removed by dissolution so as to reveal the underlying material, which can then be further manipulated. 3 The grating equation is written in a number of ways, differing in the sign of the sin terms. These alternatives simply reﬂect different sign conventions for the angles of incidence and diffraction and which side of the grating normal is allocated to þ m and m. All forms of the equation give identical results if used with the correct sign convention. Colour and the Optical Properties of Materials 206

(a)

order m = 0 (straight through) θ θi m

incident transmission grating order m beam

(b) incident beam

reflection grating

θi θ d m

order m = 0 (specular reflection)

order m

Figure 6.6 Diffraction grating: (a) diffraction from apertures in an amplitude transmission grating; (b) diffraction from reflection in an amplitude reflection grating greater intensity than those formed from a single slit. These principal maxima become sharper as the spatial frequency of the grating (the number of lines per unit length) increases. Quite ordinary gratings, with in excess of 1000 lines per millimetre, produce very narrow orders which are, in effect, imaged as lines. There are two particularly useful formulations of the grating equation. The positions of the diffraction maxima for a transmission or reflection grating when illuminated by monochromatic light normal to the surface are given by the formula:

dsinm ¼ ml 207 Colour Due to Diffraction

This is derived from Equation 6.3 by putting i ¼ 0 (Figure 6.7a and b). When light falls on a reﬂection grating at close to grazing incidence, i ¼ 90 and the formula for the positions of the diffraction maxima is:

dð1sinmÞ¼ml or

dð1cosÞ¼ml and is the angle that the diffracted beam makes with the reﬂecting grating surface, which is the complement of m (Figure 6.7c).

(a) transmission grating

θ d m incident order m = 0 beam

order m

(b) reflection grating

θ order m = 0 d m incident beam

order m

θ incident m θ order m = 0 beam d reflection grating Figure 6.7 Diffraction at normal incidence: (a) amplitude transmission grating; (b) amplitude reﬂection grating. Diffraction at grazing incidence: (c) amplitude reﬂection grating Colour and the Optical Properties of Materials 208

Figure 6.8 The continuous ﬁrst-order spectrum formed by passing the light from a pen torch through a linear transmission grating of 1000 lines/mm. The disc to the right is the zero-order spot

When a grating is illuminated by white light, each wavelength will be diffracted through a slightly different angle so that each order will consist of a spectrum except for the m ¼ 0 order, in which all of the different wavelengths overlap to give white. The resolution of a grating that is, the difference in wavelength of two adjacent lines that can be separated is a function of the spatial frequency (number of lines per unit length) of thegrating. Thegreater the spatial frequency, the greater is the resolving power of the grating. For spectroscopic purposes, especially when examining the light from faint objects such as distant stars, it is important to concentrate as much light as possible into the intense m ¼1 orders. For the simple gratings discussed, most light falls into the spectroscopically useless m ¼ 0 order. If the reﬂecting units of a reﬂection grating are cut at a slight angle to the plane of the grating, to produce a blazed grating, then the maximum intensity can be directed into any chosen spectral region. Gratings for specialist use are invariably of blazed construction. Diffraction gratings can give rise to very intense colours. Although accurately ruled gratings are expensive, plastic replicas are inexpensive and readily available. These can be used to show the spectra of many light sources, including those from torches using incandescent light bulbs (Figure 6.8) and from street lights (see Chapter 7).

6.6 Two-Dimensional Gratings

A diffraction grating consisting of a set of ruled lines is, in effect, a one-dimensional grating. The simplest two- dimensional grating is formed by two sets of ruled lines at right angles to one another. This is equivalent to an array of apertures arranged in an ordered pattern in an opaque screen. The diffraction pattern from such an arrangement consists of an array ofintense spots arranged on a grid with a symmetry that matches the symmetry of thegrating pattern. For example, a grating consisting of a rectangular array of apertures will produce a pattern consisting of a zero-order central maximum of greatest intensity surrounded by a rectangular array of bright spots. If the grating repetition is characterized by perpendicular spacings a and b then the rectangular grid of diffracted maxima will be spaced proportional to 1/a and 1/b. As before, it is important to be aware that the maxima surrounding the central peak are not the same as the weaker subsidiary maxima described above for a 209 Colour Due to Diffraction single rectangular aperture (Section 6.2 and Figure 6.4), but are new intense orders produced by the grid of the diffraction grating, as are the principal maxima in the line grating described above. For example, a net curtain acts as a (not perfectly aligned) array of square or rectangular apertures. If a far-off sodium street lamp is viewed through fairly closely woven net curtains, a square or rectangular grid of yellow spots will be seen centred upon the image of the light itself (Figure 6.9a). If the light is white, such as a beam of strongly reflected sunlight, the diffracted orders, except for the central (zeroth) spot, will be coloured, although the separation of the spectra will not begreat(Figure 6.9b). The intensity of the patterns is seen to best advantage when viewing a small, distant bright light through a closely woven black fabric such as an opened umbrella, which absorbs superfluous reflection and scattering. The array of surface pits on a compact disc makes a good reflection grating. The pits that record the data are arranged along tracks of a constant spacing (Section 3.1) on a surface that is subsequently covered with a

Figure 6.9 Two-dimensional transmission gratings: (a) a sodium streetlight viewed through a white net curtain; (b) bright sunlight reﬂected from parked cars viewed through a white net curtain. The rectangular array of diffraction orders mirrors the symmetry of the curtain mesh Colour and the Optical Properties of Materials 210

Figure 6.10 Reflection grating colours: (a) colours formed by the reflection grating on the surface of a CD; (b) colours formed by the reflection grating on the surface of a gift bag reflecting layer. The surface, therefore, forms a curved line grating and the image of a white light viewed by reflection from the surfacewill show several orders of diffraction, seen as continuous spectra, as the disc is tilted (Figure 6.10a). Because of the curvature of the tracks these spectra take a complicated form. Similar complications arise, and are put to good effect, in reflection gratings used in decorative coatings. In these, the plastic grating replicas are incorporated into the decorative pattern. The observed colours vary with viewing angle and the degree of distortion of the material (Figure 6.10b). Colour effects are often enhanced by covering the reflection grating with other coloured transparent layers.

6.7 Estimation of the Wavelength of Light by Diffraction

It is surprisingly easy to estimate the wavelength of light using a digital versatile disc (DVD), compact disc (CD), hi-ﬁ gramophone (phonograph) record or a steel rule in conjunction with the phenomenon of diffraction. The method uses these objects as reﬂection diffraction gratings, which, for a narrow beam of light, can all be 211 Colour Due to Diffraction

screen

first order

red laser pointer s1 θ m zero θ θ s0 i i order

d D Figure 6.11 The arrangement to measure the wavelength of light with a simple reflection grating such as a CD considered to be linear gratings. The spread of the diffracted spectra depends upon the spacing of the grating. The principal grating on discs is formed by the tracks, which are 1.6 mm apart on a CD and 0.74 mm on a DVD. A hi-fi gramophone record has a grating spacing of about 0.1 mm and a steel rule is graduated down to 0.5 mm. Several orders of diffraction can be seen if a narrow beam of white light from a pen torch is reflected at near to grazing incidence to a disc surface. The orders diffracted from a hi-fi record are harder to see, and those from a steel rule are the most difficult to detect. Much more accuracy can be obtained using a laser pointer. Because laser light is coherent (Section 1.9), diffraction effects are pronounced. If a laser pointer is shone on a CD at close to grazing incidence several diffracted orders will be easily visible as brightspotsonanearbywallorscreen(Figure6.11).(Ashortlineofcloselyspacedsubsidiarymaximawillappear to either side of these bright spots. They can be ignored for the present purposes, as only the most intense spot ofthisgroupistheprincipalmaximumoftheorder.)Inordertoestimatetheanglesofincidenceanddiffraction,use the fact that tan(90 i) is given by s0/D for the zero order (m ¼ 0, reflected) beam. The angle of diffraction of the order-m beam, given by tan(90 m), is given by sm/D. These values are substituted into the grating equation to obtain the laser wavelength, remembering to pay attention to the sign convention used (Section 6.5). In a home experiment, a helium neon laser pointer was attached to the top of a camera tripod by tape and the beam directed at a shallow angle onto a CD. The positions of the diffraction maxima could be measured to within 1 mm by allowing the beams to fall onto a sheet of graph paper. In a quick trial the distances were s0 ¼ 6.3 cm, s1 ¼ 23.5 cm, D ¼ 15.0 cm (Figure 6.11). Hence:

tanð90iÞ¼6:3=15:0 i ¼ 67:2 tanð90mÞ¼23:5=15:0 m ¼32:6

Substitution in Equation 6.3 with m ¼ 1 givesl ¼ 613 nm. The red laser light has awavelength of 632.8 nm. It is surprising that such an accurate value can be obtained so easily. Careful experimentation will give an answer much closer to the known wavelength.

6.8 Diffraction by Crystals and Crystal-like Structures

6.8.1 Bragg’s law

The atoms in a crystal are arranged in ordered arrays and form a three-dimensional grating. The separation of atoms in crystals is similar to the wavelength of X-rays, and the diffraction of X-rays from these Colour and the Optical Properties of Materials 212 three-dimensional gratings has been used for the elucidation of crystal structures since the early years of the twentieth century. Electrons or neutrons can also be diffracted by crystals, and both these techniques are also widely used in structure analysis. The resulting diffraction pattern is analogous to that produced by a two- dimensional grating. A grating consisting of an array of atoms placed at the corners of a lattice built up by stacking brick-like units of sides a, b and c will produce a pattern consisting of a zero-order central maximum of greatest intensity surrounded by a three-dimensional array of bright spots also arranged on a brick-like lattice. The grid of diffracted maxima will be spaced proportional to 1/a,1/b and 1/c. To be able to determine a crystal structure precisely it is necessary to measure the positions and intensities of the diffracted beams. However, even the position alone of a diffracted beam will give information about the spacing of the planes of atoms responsible. This comes about in the following way. The positionsof the diffracted beams from a line ofatoms aregivenby Equation 6.3. Focusing attention on the zero-order reflection, for conciseness, a strong diffracted beam (the ‘reflected’ beam) will occur at an angle dependent upon (but not equal to) the direction that the incident beam makes with the line of atom scatters. When the atoms are arranged on a plane, a strong diffracted beam will only occur when the diffraction maxima from each of the rows of atoms in the plane are in phase. This imposes restrictions such that a strong zero-order diffracted beam is only produced when the plane is treated as a mirror and the angle of incidence of the beam falling onto the plane is equal to the angle of ‘reflection’ of the diffracted beam off the plane. There is, though, no restriction on the angle of incidence itself. When the atom planes are stacked up to form a three-dimensional grating (i.e. a crystal), there are further limitations to the diffraction. Once again, a strong zero-order diffracted beam only occurs for ‘reflection’, i.e. when the angle of incidence on the stack of planes is equal to the angle of ‘reflection’ off the stack of planes, butin addition, only certain specific values of the spacing between the planes giverise to anysignificant intensity. This means that a strong diffracted beam only occurs at a few specific angles of incidence. Similar arguments apply to other orders of diffraction. Thus, when a crystal is bathed in a beam of X-rays, no diffracted maxima will, in general, be observed. As the crystal is tilted and rotated, sometimes a set of atomic planes in the crystal will be in just the right orientation for ‘reflection’, and the spacing of the planes will be just right for the production of a strong diffracted beam, so that an intense ‘reflection’ flashes out from the crystal, quickly being extinguished as the crystal is rotated or tilted further. The well-known formula relating the planar spacing and the occurrence of a strong diffracted beam is known as Bragg’s law. Consider the diffraction (for the reasons given above, often called ‘reflection’) of a beam, 1, of monochromatic X-rays from a plane of atoms in a crystal (Figure 6.12). Each atom acts as a point scattering centre for the

θ A B θ B d C θ B D B

Figure 6.12 The geometry of the diffraction of X-rays from a crystal lattice needed to derive Bragg’s law. Two beams, 1 and 2, are ‘reﬂected’ from two adjacent planes of atoms spaced a distance d apart and reinforce each other when the path difference between them is a whole number of wavelengths 213 Colour Due to Diffraction

X-rays and the maximum diffracted intensity will lie at the same angle B, the Bragg angle, with respect to the atom layer as the incident beam. (Care here the traditional angle of incidence used in optics, i.e. i, is the complement of the Bragg angle B). If another beam, 2, is reflected from a parallel layer of atoms a distance d below the first layer it will travel further than beam 1. For beams 1 and 2 to reinforce each other they must be in phase on leaving the crystal. (Because the process occurring with both beams is identical we can ignore any change of phase that might occur on diffraction.) This means that the path difference between beams 1 and 2 must be a whole number of wavelengths. Using the information in the figure it is seen that beam 2 has a longer path than beam 1 by CB þ BD. For reinforcement: CB þ BD ¼ ml where m is an integer and l is the wavelength of the X-rays. However:

CB ¼ BD ¼ dsinB

Hence:

ml ¼ 2dsinB ð6:4Þ where d is the separation of the planes of atoms which are responsible for the diffraction, l is the X-ray wavelength, m is the order of the diffracted beam and B is the Bragg angle between the X-ray beam and the atom planes. This relationship is Bragg’s law. It was ﬁrst applied in 1913 to determine the spacing of the lattice planes in a crystal of sodium chloride. Crystals do not scatter X-rays very strongly. Only a very small proportion of the incident beam is diffracted, and implicit in the Bragg equation is the notion that any X-ray photon is only scattered once. (This also applies to biological material, and it is for this reason that X-rays can be used in medical diagnosis.) It is not unusual to use exposure times of hours in order to obtain X-ray diffraction patterns from small crystals. The theory describing this diffraction is called the kinematical theory of X-ray diffraction. In contrast to this, electrons, which are also diffracted by crystals, interact very strongly with the atoms in a crystal. Thus, it is easy to obtain an electron diffraction pattern of a crystal in a fraction of a second, using an electron microscope (Figure 6.13). The diffraction pattern consists of an array of bright spots, each of which is derived from a plane in the crystal and conforms to Bragg’s law in position. Because the electron beam passes through the crystal, it gives information about the grating formed by atom planes parallel to the electron beam. However, verythin crystals must be used to obtainthese patterns, because each electron is scattered manytimes. If the crystal is thicker than a few nanometres the electrons are completely absorbed by the crystal and no diffraction effects are recorded. The theory describing electron diffraction for such multiple scattering is more complex than the kinematical theory and is called the dynamical theory. Moreover, because of the multiple scattering, it is not easy to relate the intensity of the scattered beams to the atomic structure of the crystal, and electron microscopy and electron diffraction are mainly used to determine crystal structures when methods such as X-ray diffraction cannot be employed. The dynamical theory of X-ray diffraction reduces to the kinematical theory when the scattering of X-rays is weak. Bragg’slaw,derivedfromthekinematical theory,isanapproximationthat sufﬁces formanyapplications.

6.8.2 Opals

The gemstone precious opal is an example of a natural material that diffracts light in the same way that ordinary crystals diffract X-rays. Common opal (potch opal) has a milky appearance and is the origin of the adjective Colour and the Optical Properties of Materials 214

Figure 6.13 Electron diffraction pattern from a single crystal of the oxide WNb12O33. The pattern is a plane section through the three-dimensional diffraction pattern. The spacing of the diffracted spots gives information on the dimensions of the crystal structure opalescent. Precious opal shows flashes of colour from within the stone, blazing out brilliantly over small angles as the stone is tilted. In the rarest opals the colours flash out from a black background. Figure 6.14 shows veins of opal in ironstone from Australia. The colours in the veins changes if the fragment is tilted only slightly. (In reality the figure does not do justice to the colours seen with an optical microscope, which reveals fleeting reds greens and blues, all of which are angle dependent and seemingly buried within the veins.) The colour of precious opal is due to the diffraction of white light. The regions producing the colours are made up of an ordered packing ofspheres of silica (SiO2) which are embedded in amorphous silica or a matrix of

Figure 6.14 Veins of opal in ironstone 215 Colour Due to Diffraction

ordered silica sphere “crystallites”

disordered spheres and amorphous silica Figure 6.15 The structure of precious opal consists of regions where spheres of silica pack together into ordered ‘crystallites’ surrounded by a disordered matrix of silica spheres and amorphous silica. The ordered ‘crystallites’ vary from one to another in orientation and in the diameters of the spheres disordered spheres (Figure 6.15). These small volumes resemble small crystallites. They interact with light because the spacing of the ordered regions of silica spheres is similar to that of the wavelength of light. These ﬂashing colours arise from regions where the spheres of silica are ordered, whereas the pale milky colour of potch opal arises in regions containing disordered spheres. 1. The conditions under which diffraction takes place are the same as those discussed with respect to the Bragg equation. However, the Bragg equation, which was derived for X-ray diffraction, must be modiﬁed in the following two ways.The incident light is not travelling in a vacuum, but in a matrix of silica, so that the diffraction conditions will relate to the wavelength of the light in the solid. Thus, use:

lðopalÞ¼l0=ns

where ns is the refractive index of the opal matrix, approximately that of silica in opal, about 1.45, and l0 is the vacuum wavelength of the light. 2. Refraction of the light beam will take place at the opal surface and the diffraction angle B in the opal will not be the same as the angle that the beam makes with the external surface. Writing 1 for the angle of incidence and 2 for the angle of refraction (Figure 6.16a):

B ¼ð902Þ Colour and the Optical Properties of Materials 216

(a) white light colour (wavelength λ) θ 1

θ 2 θ B

(b) white light red orange yellow

θ 1 yellow- surface green of opal θc

θ θ B B array of spheres

Figure 6.16 Diffraction from precious opal. (a) Diffraction from an ordered array of silica spheres in an amorphous silica matrix. (b) Not all colours will be able to escape from the opal, due to total internal reﬂection. If red light is observed normal to the diffracting layers, yellow–green light will just escape along the surface. All shorter wavelengths will remain within the opal

The form of the Bragg equation becomes:

ml0 ¼ 2nsdsinB

¼ 2nsdsinð902Þ

¼ 2nsdcos2 2 1=2 ¼ 2nsdð1sin 2Þ

Using Snel’s law:

sin1 sin2 ¼ ns 217 Colour Due to Diffraction and just writing l for the wavelength of the light observed in air: 2 1=2 l ¼ sin 1 m 2nsd 1 2 ns ð6:5Þ ¼ ð 2 2 Þ1=2 2d ns sin 1

(For precious opal this formula is adequate, but in circumstances in which the volume of the voids becomes comparable to the volume of the solid it is better to use the effective refractive index; Section 6.8.4.) When an opal is illuminated with white light some regions strongly diffract red, some green and so on, dependent upon the incident angle 1 of the incident light, the sphere diameter and the order of diffraction m. The colour seen is given by:

= ðn2 sin2 Þ1 2 l ¼ 2d s 1 m

The colour of a single grain will change with viewing angle because of the sin2 term in the equation. As the diffracting grain is tilted, the colour noted by a ﬁxed observer will move from a maximum value to lower values; that is, red colours will shift towards violet. The longest wavelength observable lmax will occur at normal incidence, when sin1 ¼ 0 and m ¼ 1. In this case the wavelength diffracted back to the viewer will be:

lmax ¼ 2nsd

At a certain angle total internal reﬂection will prevent the light from escaping (Figure 6.16b). The actual range of colour play will thus be less than that suggested by the Bragg equation. If the opal has a ﬂat surface and is surrounded by air, the critical angle c (see Chapter 2) will be given by:

1:0 sinc ¼ ns

The diffraction angle B is given by (90 c) (Figure 6.16b). Thus, it is possible to write:

lmax=lc ¼ð2 ns dÞ=ð2ns dsin BÞ¼1=sinB ¼ 1=cos c

where lc is the wavelength of the colour diffracted just at the critical angle. No light of shorter wavelength will escape. The equation for natural opal, Equation 6.5, can be generalized for light passing from a medium of refractive index n1 into a medium of refractive index n2 thus:

ml ¼ 2n2dsinð90 2Þ

¼ 2n2dcos2

2 1=2 ¼ 2n2dð1 sin 2Þ Colour and the Optical Properties of Materials 218

Using Snel’s law, sin2 ¼ n1sin1=n2 (Chapter 2): 1=2 n2 2 l ¼ 1sin 1 m 2n2d 1 2 n2 ¼ ð 2 2 2 Þ1=2 2d n2 n1sin 1

If the surrounding medium is air, n1 ¼ 1.00, and the medium of the opal is silica, n2 ¼ ns:

l ¼ ð 2 2 Þ1=2 m 2d ns sin 1 as above.

6.8.3 Artiﬁcial and inverse opals

There is considerable interest in the use of artificial opals and related structures for the diffraction of light, with the ultimate aim of employing these materials for optical data manipulation and computing. Artificial opals are generally prepared by forming polymer (frequently polystyrene or poly-methyl methacrylate) spheres in suspension and then allowing these to aggregate in controlled conditions, often aided by centrifuging. The chemistry of formation is carefully controlled so that only a small range of sizes is produced a monodisperse suspension and the solid aggregates then adopt structures analogous to those of pure metals such as copper or gold. The products are called colloidal (photonic) crystals or colloidal opals. The initial step in the formation of a colloidal crystal is the deposition of a close-packed hexagonal array of spheres onto a planar substrate (Figure 6.17a). Successive sheets of spheres with the same geometry form on top of the first, fitting into the dimples on the preceding layer. The commonest structure formed corresponds to a three-layer repeat stacking4 (Figure 6.17b). The unit cell of this arrangement is, in fact, cubic and representative of the face-centred cubic structure (Figure 6.17c). The sheets that are laid down in this way correspond to crystallographic (111) planes and the direction normal to the sheets is the [111] direction.5 In terms of the conventional crystallographic cubic unit cell, (111) planes are cell diagonal planes and the [111] direction is the cell body diagonal. For an array of cubic closest packed spheres the fraction of the volume occupied is 0.7405 and the relationship between the sphere radius and the spacing of the (111) planes is: p 2 2r d111 ¼ p 1:633r 3 It is of interest to discover that a mixture of spheres of two sizes will often aggregate to form superlattices, the structures of which are analogous to those of alloys such as brass, an alloy of copper and zinc. Inverse opals are fabricated from colloidal crystals (Figure 6.18a) by infiltrating the spaces between the polymer spheres with a suitable inorganic precursor in solution. The precursor is transformed to a solid by drying or heating. The polymer spheres that form the crystalline template are removed by solution or heating in air. The end result is a crystalline array of hollow spheres, the shells of which are made of the inorganic solid chosen (Figure 6.18b). The shells may be crystalline or amorphous, depending upon preparation methods.

4 This arrangement is called cubic closest packing. The two simplest ways of stacking the layers one on top of another are a two layer repeat called hexagonal closest packing and the three layer repeat called cubic closest packing. Many more complex packing arrangements can be devised. 5 The designation of planes in a crystal is by Miller indices (hkl). The indices h, k and l specify the fractions of the unit cell edges a, b and c intercepted by the plane. Directions [uvw] are perpendicular to (hkl) cubic crystals. Note that the type of brackets used, (xxx)or[xxx], are part of the nomenclature. 219 Colour Due to Diffraction

(a)

(b) layer A

layer B layer C

Figure 6.17 Close packing of spheres: (a) a single close-packed array of spheres; (b) cubic closest packing of spheres, with all layers identical to that in (a); (c) the cubic unit cell of the packing in (b). Each layer in (b) lies perpendicular to the unit cell diagonal, [111] Colour and the Optical Properties of Materials 220

Figure 6.18 Scanning electron micrographs of: (a) a poly-methyl methacrylate colloidal crystal; (b) an inverse opal formed of CeO2 fabricated from (a). [Reprinted with permission from Chemistry of Materials, Physical and Optical Propoerties of Inverse Opal CeO2 Photonic Crystals by Geoffrey I. N. Waterhouse et al., 20, 3, 1183-1190 Copyright 2008] 221 Colour Due to Diffraction

When these artiﬁcial structures are illuminated with white light they will strongly diffract colours of wavelength l in a similar manner to natural opals. It is convenient to replace ns in the equation for natural opal, Equation 6.5, with the effective refractive index of the opal or inverse opal phase ne (see Section 6.8.4 below) to give:

l ¼ ð 2 2 Þ1=2 m 2d ne sin 1 where m is the order of the reﬂection, d is the spacing between the layers of spheres or voids that make up the diffracting plane of the crystal and 1 is the (conventional) angle of incidence of the white light in medium 1. As the angle ofincidence increases, so thewavelength diffracted will decrease; that is, a red reﬂection at ¼ 0 will move towards green and blue, as described for natural opals above. The maximum value of l is given by m ¼ 1, due to diffraction from the (111) planes of spheres, which have the greatest value of d (see Section 6.8.5). Because of shrinkage during processing it is useful to replace the sphere radius with the more easily measured average distance between the sphere centres D:

d111 1:633r 0:8165D

The wavelength diffracted by the (111) planes is then given by:

l : ð 2 2 Þ1=2 111 1 633D ne sin 1

At normal incidence:

l111 1:633Dne

6.8.4 The effective refractive index of inverse opals

The effective refractive index ne of the opal or inverse opal can be estimated experimentally by plotting the 2 square of the wavelength diffracted from the (111) planes versus sin 1:

l2 ð: Þ2ð 2 2 Þ 111 1 633D ne sin 1 ¼ð: Þ2 2ð : Þ2 2 1 633D ne 1 633D sin 1

The slope and intercept of the linear graph (Figure 6.19a) are given by:

slope ¼ð1:633DÞ2 ¼ð : Þ2 2 intercept 1 633D ne

The effective refractive index can be related to the refractive indices of the components of the inverse opal in several ways. The most widely used is to employ the volume fractions (Section 2.5) thus:

ne ¼ n1V1 þ n2V2 þ n3V3 þ

V1 þ V2 þ V3 þ ¼1 Colour and the Optical Properties of Materials 222

(a) intercept 2 (1.633 D) ne

λ 2 111

slope – (1.633 D)2

2 θ sin 1

(b)

λ 111

intercept (1.633 D) n1 V1 slope (1.633 D) n2 (1 – V1)

n2 l2 2u l l Figure 6.19 Schematic plots of (a) 111 versus sin 1; (b) 111 versus n2 for an inverse opal structure; 111 u is the wavelength of light diffracted by the (111) planes of the array, 1 is the angle between the normal to the surface and the incident beam of white light and n2 is the refractive index of the ﬂuid ﬁlling the voids in the structure

where ni is the refractive index of the ith component and Vi is the fraction of the total volume of the solid occupied by the ith component. For a two-component system, such as that composed of spheres and air or spherical shells and air:

V2 ¼ 1V1

If the air-ﬁlled voids in an inverse opal in air are ﬁlled with a liquid, the effective refractive index change will result in a change in the colours diffracted. Generally speaking, the wavelength diffracted increases, so that, for example, blue green diffracted beams become red on adding liquid. For simplicity, consider an inverse opal in air diffracting light from the (111) planes. At normal incidence: 223 Colour Due to Diffraction

l111 ¼ 1:633Dne

ne ¼ n1V1 þ n2ð1V1Þ where subscript ‘1’ refers to the walls and subscript ‘2’ to the voids. Thus:

l111 ¼ 1:633D½n1V1 þ n2ð1V1Þ

¼ 1:633Dn1V1 þ 1:633Dn2ð1V1Þ where n2 is the refractive index of the liquid within the void. Using a series of different liquids, a plot of l111 versus n2 (Figure 6.19b) will be a straight line with the parameters:

slope ¼ 1:633Dn2ð1V1Þ

intercept ¼ 1:633Dn1V1

Once this graph has been constructed, the inverse opal can be used as a refractive index meter for the determination of an unknown refractive index.

6.8.5 Photonic crystals and photonic band gaps

Photonic crystals are artificial structures that have unit cells with dimensions approximately equal to the wavelength of light and so give rise to intense diffraction colours. The ‘crystal’ can be made up of arrays of particles (as in opals), voids (as in inverse opals) or any other structures (tubes, layers and so on) provided that the structural repeat distance is similar to the wavelength of light. In addition to structures fabricated in the laboratory, many beautiful colours in nature are produced by natural photonic crystals created by living organisms. Probably best known are some of the spectacular colours of certain butterflies or beetles, but lesser known creatures, such as the sea mouse, have vividly coloured spines due to a photonic crystal type of microstructure (see this chapter’s Further Reading). There is considerable interest in the fabrication of photonic crystals, including artificially mimicking the natural photonic crystals that appear in living organisms, because these provide a compact way of manipulating photons without additional energy requirements. Because of this research perspective, the terminology used in electronics has been used to describe some aspects of the physical processes that occur on reflection and diffraction. Thus, when a colour is strongly reflected by an opal or an opal-like array it will not pass through the solid. In the jargon of photonics, this state of affairs corresponds to a photonic band gap (PBG) or a stop band. Thus, a PBG or stop band occurs when a range of frequencies will not propagate through the crystal. The terms reflection, Bragg reflection, stop band and PBG are frequently used synonymously. More explicitly, a complete PBG occurs when the propagation of a range of frequencies is forbidden for every state of polarisation and propagation direction. The rules regarding the existence of PBGs are identical to those that determine if an X-ray beam will be diffracted and are documented in crystallography texts (see this chapter’s Further Reading). The positions of the band gaps, corresponding to the strongly diffracted wavelengths, are readily computed via Bragg’s law, which holds for any ‘crystalline’ array no matter the size of the constituent ‘atoms’ and whether man-made or not. As an example, the situation in the type of colloidal crystal described above will be outlined. These colloidal crystals and inverse opals are laid down in sheets of (111) planes described in terms of a cubic unit cell. The strongly diffracted wavelengths are given by Bragg’s law, and it remains to calculate the interplanar spacing Colour and the Optical Properties of Materials 224

(d values) of the various planes of spheres or voids making up the solid. For the close-packed arrangement described, the interplanar spacing is given by:

a dhkl ¼ p h2 þ k2 þ l2 where a is the cubic lattice parameter. The relationship between the measured average distance between the sphere centres D and the cubic lattice parameter a is: p a ¼ 2D Hence: p 2D dhkl ¼ p h2 þ k2 þ l2 There is one other factor to take into account. Not every plane in the structure will give rise to a strongly diffracted beam. This is because, for some specific values of (hkl), interference effects not described previously cancel out the beams diffracted from adjacent planes to give zero diffracted intensity. For example, there is no Bragg reflection from a (100) plane in a face-centred array of the same arrangement as found in artificial opals. The lowest order reflecting planes (hkl) for this structure in order of interplanar spacing are (111), (200), (220), (311) and (222).6 PBGs will then occur for each of these planes. The wavelength and angle dependence is given by the equations above, simply by substituting the appropriate value of dhkl; that is: q l ¼ 2 2 m 2dhkl ne sin 1 p q 2 2D 2 2 ¼ p n sin 1 h2 þ k2 þ l2 e

For illumination perpendicular to the planes: p 2 2Dn ml ¼ p e h2 þ k2 þ l2

If the photonic crystal is thin, then the transmitted colour will be complementary to the reflected colour. The production of these different reflected and transmitted colours is sometimes called optical filtering. (For more information on the enormous topic of photonic crystals, see this chapter’s Further Reading.)

6.8.6 Dynamical form of Bragg’s law

In the foregoing section the kinematical (single scattering event) theory of scattering is used. However, light photons are strongly diffracted by gratings and very little light would penetrate a stack of gratings. This means

6 Note that reflection from (200) corresponds to the second order m 2 reflection from (100), the reflection from (220) corresponds to the second order reflection from (110) and reflection from (222) corresponds to the second order reflection from (111). X ray crystallography has adopted the system of keeping m 1 and changing the interplanar spacing as in the equation for dhkl given. In optics it is more common to keep the diffraction plane as constant and vary the order m. In either case the numerical results from both approaches are exactly the same. 225 Colour Due to Diffraction that for a precise understanding of the scattering by opals and similar arrays the dynamical (multiple scattering) theory is required. For normal incidence, the dynamical theory gives Bragg’s law as: j l ¼ l 1 þ d B 2 where ld is the wavelength of the diffraction peak maximum computed by dynamical theory, lB is the Bragg wavelength and j is related to the ratio of the refractive indices of the wall nwall and voids nvoid in the following way:

n2 1 j ¼ 3V rel wall 2 nrel 2 nwall nrel ¼ nvoid

For the instance of diffraction normal to (111) planes in air: j l ¼ ½1:633Dn V þ 1:633Dn ð1V Þ 1 þ d wall wall void wall 2

And in general, for normal incidence:

l ¼ 2 m d 2dne j ¼ 2dn½V þ n ð1V Þ 1 þ wall wall void wall 2

These corrections are needed in the most precise work.

6.9 Diffraction from Disordered Gratings

6.9.1 Random specks and droplets

Randomly sited copies of a single object will produce a diffraction pattern which is a brighter version of that of the isolated object. Thus, the diffraction pattern of a random collection of circular apertures or rectangles will consist of the same patterns as described above, but with an increased intensity. This has interesting consequences for pattern recognition. For example, suppose that the object consists of an array of pairs of circular apertures arranged so that the axes of the pairs are parallel but the position of the pairs is at random over a plane. To the eye this will resemble a random collection of single circular apertures. The diffraction pattern will not, however, look like an Airy pattern, but will be a fringe pattern consistent with that from a single pair of apertures with the fringes perpendicular to the axis of the aperture pair (Figure 6.20a and b). If the pairs are arranged randomly both with respect to position and the orientation, the single fringe pattern will be duplicated at every angle that the pairs of points display in the object. With enough point pairs the unidirectional fringes are transformed into a diffuse ring pattern (Figure 6.20c and d). This provides a very simple way of distinguishing between a random array of objects and an array that appears to be random but does include some hidden order. For example, the diffraction pattern of an amorphous material, such as a glass or a ﬁlm of evaporated carbon, usually consists of a few very diffuse rings (Figure 6.21), indicating that a certain degree of order is present, although invisible to the eye. If the atoms were truly independent of each other, the diffraction pattern would be Colour and the Optical Properties of Materials 226

(a) (b)

1/d

Figure 6.20 Diffraction patterns of arrays of points (schematic). (a) A random array of pairs of points (one pair in box, separation d). (b) Diffraction pattern of (a) consists of fringes of separation 1/d. (c) A random array of pairs of points as in (a), but with random orientation. (d) Diffraction pattern of (c) consists of a set of diffuse rings of spacing 1/d. By eye, neither (a) nor (c) indicate the presence of any internal order, but the diffraction patterns show this clearly an Airy pattern equivalent to that from a random collection of points. In a silicate glass, for example, the diffuse rings reflect the occurrence of many Si O bonds, each of which have a similar length but which are arranged at random with respect to the incident illumination. In carbon films the C C bond plays the same role. The average bond length in the amorphous state can be estimated from the ring diameter and compared with bond distances in crystalline materials. Diffraction patterns from random droplets or specks can be seen frequently. Because of the wavelength sensitivity of the diffraction, the effects give rise to colours. One of the commonest of these phenomena is the corona around the sun or moon, seen through high, thin clouds.7 They lie close to the disc of the object and are much narrower than the halos described earlier (Section 2.8). The pattern is the Airy ring (Fraunhofer) diffraction pattern from the collection of randomly distributed droplets. These add together and an observer, in reality, sees fragments of the diffraction patterns from many droplets or specks, each of which contributes to the overall effect. When the clouds consist of similarly sized droplets or specks, the effect will be strong. At their best, the coronae show multicoloured rings surrounding the central disc of the sun or moon. Usually only the first ring is easily seen, and has a colour sequence with violet on the inside and red on the outside. If more than one ring is possible then the same colour sequence, violet inside and red outside, is seen. When the drops are variable in size the effect is diminished and then just a pale ring can be made out. For a similar reason, a multicoloured ring can sometimes be seen to surround a narrow beam of white light which has passed through a pane of glass covered with a fine powder or with fine drops of moisture. Each particle diffracts as a small circular aperture. The eye intercepts many of these diffracted rays and a coloured

7 This is different from the outer atmosphere of the sun, which is also called the corona. 227 Colour Due to Diffraction

Figure 6.21 Amorphous carbon film: (a) high resolution electron micrograph; (b) diffraction pattern. The diffuse rings indicate that some order is present in the apparently random film. [Reprinted with permission from Geoffrey I. N. Waterhouse et al., Physical and Optical Properties of Inverse Opal CeO2 Photonic Crystals, 20, 3, 2008, American Chemical Society] ring is seen which is composed offragments of colour from many different dust particles. As before, violet is on the inside and red is on the outside of the circle. The same effect can sometimes be seen around the image of a small light in a dusty mirror. Again, each dust particle acts so as to diffract the light, which is reflected from the mirror surface back towards the observer and is the Fraunhofer diffraction pattern of the specks. Another series of coloured fringes can also arise from dust on a mirror surface. In this case the ring pattern, which is also multicoloured, is called the Whewell Quetalet pattern. The ring structure is again caused by diffraction from light scattered from the particles reflected from the mirror and then returned to the eye, but the interference paths of the rays are different from those that give rise to the Fraunhofer pattern. Colour and the Optical Properties of Materials 228

6.9.2 Colour from cholesteric liquid crystals

The structure of a liquid crystal made up of long molecules (a nematic liquid crystal) was described in Section 4.13. In these mesophases, the director, which is the average direction taken by the long axes of the molecules, falls along just one single direction. In cholesteric phases, which are also called twisted nematic phases or, more correctly, chiral nematic phases, the director rotates steadily as one travels along a direction perpendicular to the sheets of molecules (Figure 6.22). The result is the generation of a helical structure within the material. The physical reason for the rotation is that the molecules in each layer are asymmetric. When such molecules pack together the interactions are minimised if molecules in one layer rotate slightly compared with those in the layer below. Because these interactions are the same from one layer to another the same twist occurs between each layer. In this way the uniform helical structure results. The helix so formed can be right or left handed, and this inﬂuences the way in which a beam of white light interacts with the mesophase. Unpolarised light can be regarded as two beams of oppositely circularly polarised light (Section 4.1). The beam with the same handedness as the helical arrangement of the mesophase passes straight through an ordered or partly ordered array, while the opposite beam may interact with the array and a coloured diffracted beam can then appear. This colour arises by diffraction when the pitch of the helices in the cholesteric mesophase (that is, the repeat distance along each helix) is similar to the wavelength of light. Scattered light can then interfere constructively.

axis

director

Figure 6.22 The cholesteric liquid-crystal structure. The average orientation of the molecules (the director) in each layer rotates in a regular fashion as one moves along the axis to create a helical structure 229 Colour Due to Diffraction

The conditions for this to occur follow Bragg’s law:

ml ¼ 2nmdsinB where nm is the average refractive index of the mesophase, d is the helical repeat distance and B the angle between the layer and the light beam (Figure 6.23). (Corrections for refraction at the surface of the cholesteric phase can also be made, if important.) For light normally incident on the ﬁlm:

l ¼ 2nmd

When illuminated with white light, any wavelength satisfying the Bragg relationship will be diffracted strongly and give a colour in reﬂection. If the liquid crystal is backed by a dark background then the colour will appear quite bright. This is because the transmitted light will be absorbed. On viewing the ﬁlm normally and then moving towards grazing incidence the colour will appear to change towards shorter wavelengths due to the sin term in the Bragg equation. Because of the uniaxial nature of the molecules, the refractive index of the

white light

θ B

Figure 6.23 Diffraction of light from a cholesteric chiral nematic array. The pitch d of the array determines the wavelength of light that is strongly diffracted Colour and the Optical Properties of Materials 230 medium will be different along and perpendicular to the molecular axis. The wavelength range diffracted will be given by:

Dl ¼ 2Dnd where Dn is the birefringence of the molecules (the difference between the refractive indices along the two axes of the molecule). The pitch of the helical structure can be engineered by both temperature and impurities. The end result is widely seen in liquid-crystal thermometers. A commonly encountered form of these inexpensive devices consists of a card with a black strip of plastic running across it. In the black band a coloured number will be seen corresponding to the temperature. A different number lights up at a different point on the band as the temperature varies. These devices operate in the following way. A series of spots of a cholesteric material are arranged in a row. They are chosen so that the periodicity of the molecular helix in each spot will diffract visible light at a precisely defined temperature. An increment of the temperature dT of the order of 1 C is engineered between each successive spot by additives or by slightly modifying the cholesteric molecule. Within the design temperature range, each spot will diffract light only when the mesophase is at the correct temperature and not do so otherwise. This causes the appropriate temperature value to ‘light up’. Moreover, on approaching the temperature, each spot will run through a spectrum of colours as the pitch of the helix varies. The effect of temperature on colour, of which this is an example, is called thermochromism. These effects have already been anticipated by nature, and a number of beetles show iridescent colours due to a cholesteric arrangement of layers of fibres in the outer integument of the body. The fibre direction in each layer is slightly different from that on either side and a helical layered structure is built up. If the pitch of the spiral is of similar dimensions to the wavelength of light then they give rise to intense ‘metallic’ colours when viewed in white light. One of the puzzles surrounding early research on cholesteric liquid-crystal phases centred upon the fact that these often showed a transient bright blue colour on cooling. The effect was confined to a narrow temperature range close to the upper melting point of the phase, where the true liquid forms. This transient state contains several different forms and they are known as the cholesteric blue phases. The blue colour arises by diffraction of white light from a superstructure within the mesophase. These superstructures consist of ordering of the cholesteric helices into supercells with a dimension of the same order of magnitude as blue light. They act as crystals and diffract blue light in accordance with Bragg’s law. The ordering is only stable over a narrow range of conditions, which accounts for the fleeting nature of the feature.

6.9.3 Disordered two- and three-dimensional gratings

The situation described for a random array of specks or droplets applies equally well to random arrays of two- or three-dimensional gratings. A scattering object which is made up of a random collection of two- or three- dimensional gratings will give a diffraction pattern which is a brighter version of that of the isolated object. Here, there are two extravariables to consider besides the random spatial position: the relative orientation of the grating fragments (that is, the relative rotation about an axis parallel to the illuminating light beam) and the physical extent of the grating. Both of these modify the diffraction pattern observed. The formation of iridescent suspensions of polymer spheres provides an example. During the preparation of colloidal crystallites, monodisperse suspensions of polymer can order whilst still in the ﬂuid, well before solidiﬁcation occurs. The extent of the ordering will depend upon the concentration of spheres in the suspension and the interaction between them. A frequent form of ordering involves the spheres aggregating into hexagonally packed layers within the liquid phase. These disordered two-dimensional gratings may diffract 231 Colour Due to Diffraction white light and give rise to intense colours. When the concentrations and interactions are favourable, three- dimensional ordering can also occur over small volumes of the suspension, giving rise to diffracted colours when the sphere spacing is appropriate. Diffraction by disordered three-dimensional structures is most commonly described with respect to the X-ray diffraction pattern from a large number of randomly oriented crystallites the ‘powder method’ of X-ray diffraction. As described above, a small crystal will give rise to a strong diffracted beam when the interplanar spacing and the angle of incidence of the X-ray beam agree with Bragg’s law. The diffraction pattern from a single crystal will consist of a three-dimensional array of spots with a spacing and symmetry that matches the dimensions of the atomic grating that makes up the crystal. If the crystallites are randomly oriented, the diffraction pattern from each one will also be randomly oriented. Each spot in a single crystal pattern can now form anywhere on the surface of a sphere and the spot pattern becomes a set of concentric shells. A plane section through the pattern yields a series of rings, typifying a powder X-ray diffraction photograph. If the number of crystallites included in the incident beam is rather small, the rings are broken up into ‘spotty’ rings (Figure 6.24). The effect of limited crystallite size is to broaden the extent of each diffraction spot. In the case of a ring pattern, each ring will become broadened. This is analogous to the effect described earlier, in which a small aperture produces a greater spread of diffracted intensity than a wide aperture. The spread of the rings can be used as a measure of crystallite size.

6.10 Diffraction by Sub-Wavelength Structures

6.10.1 Diffraction by moth-eye antireﬂection structures

Night-ﬂying insects need to optimize the amount of light that reaches the receptor cells in the eye. This has been achieved by covering the surface of each of the components of the insect compound eye (the ommatidia) with a large number of tiny bumps that are somewhat smaller than about half the wavelength of the incident light, being about 200 nm at the base and 200 nm high (Figure 3.15). This creates an AR coating and these types of

Figure 6.24 A ‘spotty’ electron diffraction pattern from a polycrystalline sample of titanium dioxide (TiO2) Colour and the Optical Properties of Materials 232 surface AR coatings are also called moth-eye AR coatings. The description of the optical consequences of these surface features can be made in terms of GRIN effects (Section 3.7) or in terms of a two-dimensional diffraction grating, i.e. an ultrahigh spatial-frequency surface relief grating. This latter aspect is described here. The moth-eye surface grating is a reflection grating. To optimize night vision, it is important to minimize all reflected light. This means that the diffracted orders must be suppressed. Although the surface grating is two- dimensional, it is possible to gain an idea of the action of the small surface bumps on the surface of the moth eye by using the one-dimensional diffraction grating equations given above. Consider, initially, light prevented from entering the eye by diffraction from the reflection grating on the surface. For light falling on the surface at normal incidence:

dsinm ¼ ml where d is the grating spacing, m is the order of the principal maximum, l is the wavelength of the light and m is the diffraction angle of the mth-order maximum. The equation shows that, as the value of d approaches l, the diffracted orders make larger angles with the surface normal. The limiting cases occur when sinm is equal to 1 and m ¼ 1. Inserting these values into the equation shows that the limit is reached when l/d ¼ 1. That is, the grating spacing corresponds to the wavelength of light. All diffracted beams except the zero (m ¼ 0) order will be suppressed when d is slightly greater than l, because that would correspond to a value of sinm > 1. At the other extreme, consider the light which hits the eye surface at grazing incidence. The equation for diffraction is now:

dð1cosÞ¼ml

As before, the limiting cases occur when cos is equal to 1 and m ¼ 1. Inserting these values into the equation shows that the limit is reached when l/d ¼ 2. Thus, in either case, if the grating spacing is less than l/2, all diffracted orders will be suppressed except the zero (m ¼ 0) order. The light entering the eye will be maximised if the surface has a grating with a repeat of about l/2. For blue light this corresponds to approximately 200 nm. In order to determine the intensity of this zero-order reflection for a moth-eye structure it is necessary to calculate the intensity as a function of the surface profile (which may be square, sinusoidal or irregular), the depth of the grooves, the angle of incidence and the polarisation of the light. Generally, the calculations are numerical and no analytic solutions are available (see this chapter’s Further Reading). The diffraction equation can be generalized for application in other similar situations; for example, a fish eye adapted to dim light conditions, which must also optimise light gathering. When light passes from a medium with refractive index n1 into a medium with a refractive index n2 (Figure 6.25), the grating equation is:

ml ¼ dðn1sini þ n2sinmÞ where n1 is the refractive index of the medium containing the incident beam (above the grating) and n2 is the refractive index of the medium containing the diffracted beam of order m (below the grating). In the case of a reﬂection grating, n1 ¼ n2 and the grating equation is:

ml ¼ dðn1sini þ n1sinmÞ ¼ n1dðsini þ sinmÞ

When the initial medium is air, with a refractiveindexof unity, the equation reduces to Equation 6.3. Clearly, the same reasoning used for moth eyes applies, except now the factor n1 must be taken into account. The limiting 233 Colour Due to Diffraction

incident beam

m = 1 m = 0

θ r1 reflection

θ i m = –1

refractive index n1

d refractive index n2

θ t1

m = –1

m = 1 m = 0

transmission

Figure 6.25 Diffraction at the surface between two transparent media of refractive index n1 and n2. The angle of u u incidence is i, the angle of diffraction for the first-order reflected ray is r1 and the angle of diffraction of the first- u order transmitted ray is t1 spacing is then l/n1 for normal incidence and l/2n1 for grazing incidence. That is, the critical spacing is reduced by a factor of n1.

6.10.2 The cornea of the eye

The cornea of the eye provides a second example of diffraction by structures with a repeat dimension less than the average wavelength of visible light. The cornea is the outer surface of the eye directly in front of the lens and iris and has a thickness of approximately 0.6 mm. Naturally, it is vital to survival for the cornea to be completely transparent. Now the cornea is made up of long collagen fibrils embedded in a gel-like transparent protein. The refractive index of the fibrils is close to 2.17, while the surrounding matrix has a refractive index of 1.81. This arrangement seems ideal for giving rise to considerable incoherent Mie or Rayleigh scattering, thus rendering the cornea opaque like the rest of the ‘white’ of the eye. Transparency is obtained by making use of coherent scattering, that is diffraction, from a carefully constructed arrangement of layers, to form a transmission grating that is able to eliminate all diffracted orders except the directly transmitted beam. Apart from an exterior and interior film, the epithelium and endothelium, the cornea consists of about 250 lamellae stacked one on top of the other (Figure 6.26). Each lamella contains long collagen fibrils of a diameter of approximately 30 nm arranged parallel to each other and stacked up in an ordered array with about 50 nm between centres, into a lamella 2400 nm thick. The long collagen molecules are birefringent and so each lamella is also birefringent. However, the direction of the collagen fibrils changes abruptly from one lamella to the next. In fact, the lamellae are distributed so that the collagen molecules are arranged in a cross-ply fashion, so that over the 250 or so lamellae the effect is averaged and the cornea is not able to detect polarised light. Colour and the Optical Properties of Materials 234

stroma ~250 lamellae ~2.4 μm thick epithelium endothelium

light

light fibril

1 lamella ~2.4 μm

Figure 6.26 The cornea of the eye (schematic). (a) The main part of the cornea, the stroma, is composed of about 250 lamellae. (b) Three lamellae, each containing collagen ﬁbrils approximately 30 nm apart in a quite well- ordered array. Those at the left and right are roughly parallel to the page and those in the middle are roughly perpendicular to the page

The arrangement of the ﬁbrils on a regular grid of spacing approximately 50 nm will give rise to sub- wavelength diffraction. According to the grating equation, Equation 6.3, diffraction will occur when:

dsinm ¼ ml

In the present case, d is of the order of 50 nm, i.e. about 0.1l, and the only real solution is when m and m are both equal to zero. In essence, the scattering is coherent and reinforcement occurs for the ‘straight through’ beam while all other scattering is suppressed by destructive interference.

6.10.3 Some blue feathers

As has been apparent throughout this chapter, many of the intense colours found in nature are produced by diffraction. Sub-wavelength diffraction has also been found to be important here. A recent example is given by the iridescent blue colour of the Plum-throated Cotinga, Cotinga maynana. The colour arises in the feather barbs. Rather like the cornea, the barbs are made of a mixture of substances, in this case keratin separated by air spaces. Keratin is a tough fibrous protein widely distributed throughout the animal kingdom, occurring, for example, in hair, hooves, feathers and fingernails. Again, this structure would be expected to scatter light strongly. This is averted because the tissue is not a random arrangement, but consists of a disordered diffraction grating (Section 6.9.3). The (computed) 235 Colour Due to Diffraction diffraction pattern confirms this. The grating spacing, averaging about 165 nm, is below the wavelength of light and ‘normal’ diffraction is not possible. Bragg’s law, Equation 6.4:

ml ¼ 2d sinB suggests that, for normal incidence and m ¼ 1, the colour strongly diffracted would be 330 nm, in the ultraviolet. This estimate is too simplistic by far, but exact calculations give a reﬂectance peak in the 500 520 nm range, in good agreement with measurement. Once again, the structural colour is produced by coherent scattering from a partly ordered matrix with sub-wavelength repetition. The desirable coloured reﬂected light is enhanced by constructive interference and other colours are lost by destructive interference, arising in the ordering of the keratin air barb microstructure.

6.11 Holograms

6.11.1 Holograms and interference patterns

Objects are perceived when a train of light waves enters the eye and the resulting nerve impulses are processed by the brain. Strictly speaking, the source of the waves entering the eye is unimportant for the perception to occur. A hologram is a permanent store of the information needed to create these light waves in such detail that the observer is given the impression that the real object is being observed even though it is, in fact, an illusion. There are many forms of holograms. In this section, only those that really involve colour production are described. This chapter’s Further Reading will give more information on broader aspects of the subject. The information stored is an interference pattern formed between light scattered from an object and light that has not been scattered. The most important requirement is that the object should be illuminated by coherent light and the interference pattern is created by an overlapping of the reﬂected light with the unchanged incident beam. To do this, a monochromatic laser beam with a wavelength somewhere in the visible is divided into two. One part illuminates an object and some of the light is reﬂected from the object to create the object beam or signal beam. The other part of the beam traverses an identical distance but without having encountered the object, to form the reference beam. The object and reference beams are arranged to intersect and in this region the two reunited beams will interfere with each other and an interference pattern of variable irradiance and phase will be present. Holograms are recorded versions of these two- or three-dimensional interference patterns.

6.11.2 Transmission holograms

A transmission hologram is formed when the object beam and the reference beam enter the recording medium from the same side (Figure 6.27a). In order for the wave pattern to record the image details accurately, all vibrations must be eliminated. Any disturbance at all will introduce additional ‘information’ into the interference pattern, which amounts to a degradation of the record. To view the holographic image, a procedure known as image reconstruction, the recorded hologram is illuminated with the reference beam alone (Figure 6.27b). The beam passes through the hologram, which is why these holograms are called transmission holograms. If the viewer looks back along the reference beam a reconstructed virtual image of the object is seen, as if looking through a window, which is the hologram frame. As the viewer moves, the image remains ﬁxed in space, but the aspect of the image that is seen and its position within the hologram frame changes. The image appears to show all of the three-dimensional properties of the original object. (Exactly the same impression is created when a real object, a tree, say, is viewed through a Colour and the Optical Properties of Materials 236

(a) laser beam

spatial filter beam splitter mirror

object spatial filter

hologram recorded here mirror

(b) laser beam

spatial filter beam splitter mirror

(virtual) image

hologram “window”

Figure 6.27 Transmission holograms: (a) recording the hologram; (b) reconstruction of the image; (c) reconstruction with a different wavelength giving rise to a displaced and distorted image window. As one moves around the room, the tree stays in the same place, but the extent of the view of the tree is curtailed by the window frame.) In point of fact, the reconstruction results in two images, a real image formed in front of the hologram (i.e. between theviewer and the hologram) and avirtual image, just described, formed behind the hologram. The real image is rather difﬁcult to locate and the image most often viewed is the virtual image seen when looking into the hologram. The way in which the holographic record is formed, by the two interfering wave fronts approaching the recording medium from the same side, results in the interference pattern being restricted to a thin layer. This 237 Colour Due to Diffraction

c laser beam

spatial filter beam splitter mirror

(virtual) image

hologram

Figure 6.27 (Continued) type of hologram is called a thin or planar hologram. If such a hologram is illuminated with laser light of a different colour, then an imagewill still be formed. However, the imagewill be distorted and displaced, so that if the laser light is red, when green light is used to create the hologram, the image will appear to the left of the original because red light is diffracted more than green (Figure 6.27c). What is more, the image will move around disconcertingly as the viewpoint is changed. Similarly, if violet light is used, the image will form to the right of the original image, as violet light is diffracted less than green, and the reconstruction will be distorted and move as the viewer’s viewpoint changes. If white light is used to reconstruct the hologram the result will be a blurred and distorted shape that may not be recognizable. This type of hologram, therefore, is not suited to white-light reconstruction. The behaviour of a thin hologram can thus be seen to parallel that of a thin diffraction grating. On irradiation with white light a thin hologram givesrise toseveral orders of diffracted light, of decreasing intensity as the diffraction angle increases.

6.11.3 Reﬂection holograms

A reflection hologram is formed when the reference beam and the object beam enter the recording medium from opposite sides (Figure 6.28a). Although vibrations must be eliminated to record a good hologram, the stringent requirements required for a transmission hologram can be considerably reduced in a reflection hologram by placing the object more or less in contact with the recording medium. Image reconstruction is similar to that for a transmission hologram. However, in this case the beam is reflected from the hologram and does not pass though it (Figure 6.28b). Because the reflection holographic record is formed by the two interfering wave fronts approaching the recording medium from opposite sides, the interference pattern extends for a considerable distance through the recoding medium. This type of hologram is called a thick or volume hologram. There is a significantdifference between reconstruction using a reflection (volume) hologram compared with using a transmission (planar) hologram. In this case of a volume hologram, if the hologram is illuminated with Colour and the Optical Properties of Materials 238

(a) laser beam

spatial filter beam splitter mirror

object spatial filter hologram recorded here

mirror

(b)

laser beam or white light

hologram

(virtual) image

Figure 6.28 Reflection holograms: (a) recording the hologram; (b) reconstruction of the image with the same laser light or white light gives a coloured image white light, the interference pattern in the hologram is able to interact selectively with light of the appropriate wavelength and so a coloured reconstructed image is still visible. The difference between a planar and volume hologram is, in point of fact, analogous to the difference between a planar diffraction gating and a three- dimensionalgrating, such as a crystal (Sections 6.5, 6.6 and 6.8). The selectivelycoloured reconstruction from a volume hologram is described as arising from ‘Bragg planes’ in the holographic record. The comparison is quite accurate. When a beam of‘white’ X-rays (that is, an X-ray beam with a spreadof wavelengths) illuminates a crystal, only those wavelengths that fit the Bragg condition will be diffracted by the three-dimensional distribution of electron density within the crystal. In the same way, when a beam of white light interacts with a thick hologram, only those wavelengths that fit with the ‘Bragg condition’ imposed by the variation in interference fringes within the volume of the hologram will be diffracted and so contribute to the reconstructed image. In comparison with a thin hologram and a thin diffraction grating, a volume hologram gives rise to one diffraction order (in addition to the zeroth order), which is intense in a direction satisfying Bragg’s law. This leads to another point of interest, but one which is not restricted to volume holograms alone, but also applies to planar holograms. Each fragment of the hologram contains sufficient information to reconstruct the 239 Colour Due to Diffraction image. That is, if a hologram is smashed into fragments, each will serve to reconstruct the image. This is the same as in X-ray diffraction, where a large crystal can be continually subdivided but the diffraction pattern still contains all the information to deduce the structure of the diffracting crystal. Of course there are limits. As the crystal volume decreases beyond a certain point, information is lost and the crystal structure so deduced may be incomplete. Similarly, if the hologram fragment becomes too small, information is lost and the reconstructed image degrades.

6.11.4 Rainbow holograms

Rainbow holograms are the brightly coloured holograms seen on credit cards, security documents and so on. The first point to be clear about is that rainbow holograms are transmission holograms, even though they give the appearance of being reflection holograms. In reality, the white viewing light passes through the hologram and is reflected from a backing layer before reaching the observer. The image seen and the colours visible depend upon both the viewing angle and the viewing distance, so that a multiplicity of colours and patterns can be picked out as the hologram is tilted and moved. They were invented by Benton in 1968, and are also known as Benton holograms. Technically they are described as white-light transfer transmission holograms. In order to understand this terminology it is necessary to consider transmission hologram construction. The shortcomings which prevent transmission holograms being viewed satisfactorily in white light were overcome by two innovations. Take the problem of colour spread and distortion first. The degree of distortion and movement of an image formed by light of a different wavelength to that of the original laser depends upon the distance d between the hologram plane and the object (Figure 6.29a). If d is reduced to zero, some distortion will still exist, but the different coloured reconstructions will overlap and, as the viewpoint is moved, image movement will be minimal. Naturally, the object cannot be placed right up against the recording medium because it would block the reference beam. This is surmounted by making a hologram of the reconstruction. In this case it is necessary to use the real image formed, not the virtual image normally viewed, and this is positioned in the recording plane of the second hologram (Figure 6.29b). The original hologram is called the master, or H1, and the second hologram the transfer hologram, or H2. When H2 is viewed in white light the multicoloured (virtual image) reconstructions now overlap and do notmove much when the observation point is moved (Figure 6.29c). The reconstruction may be variously coloured or it may appear as black and white, depending upon the colour overlap and the nature of the illumination. The amount of the reconstructionvisible to the observer is, however, constrained. The transfer hologram, H2, is a hologram of a hologram, and so the observer would see the reconstructed image as if it were viewed through the same ‘window’ or ‘porthole’ that circumscribed the original hologram H1 (Figure 6.29d). If theviewpoint is changed too much, parts of the imagewill seem to fall behind this aperture and so be blocked for the observer. Moreover, each wavelength of light used in the reconstruction will create its own window, with red corresponding tothe most diffracted light and violet the least. Thus, an observeron a levelwith the hologram will see a green reconstruction, whereas if the observer moves up the image colour will move through yellow and orange to red, or if the observer moves down to green, then blue, indigo and violet (Figure 6.29e). There will still be considerable overlap of colours in parts of the image, dependent upon the viewing distance and angle. This latter problem is overcome by making the viewing window a narrow strip. H1, therefore, takes the form of a narrow strip rather than a rectangular or square frame. This time, as the observer moves up, the part of the reconstruction visible through the narrow window changes colour from green to yellow, orange and red. Similarly, on moving down, the part of the reconstruction visible to the observer changes colour sequentially from green to blue, indigo and violet (Figure 6.29f). The result is a rainbow hologram. There is a price to pay for this, because the strong sensation of depth in a ‘vertical’ direction, normal to the slit length, is lost. This is not so along the strip, in a ‘horizontal’ direction, and here, as the observer moves to and fro, Colour and the Optical Properties of Materials 240

(a) laser beam

d object hologram H1

(b)

laser beam laser beam

real image

H1 H2

H2 Figure 6.29 Transfer transmission holograms. (a) The distortion and movement of white light images in a conventional transmission hologram H1 increases as the object distance d increases. (b) A transfer hologram H2 is made from the master H1 transmission hologram with the real H1 image in the plane of H2. (c) Illumination of H2 with white light reconstructs coloured virtual images that overlap (exaggerated here) and show minimal distortion and movement. (d) The image seen will still appear as if viewed through the H1 ‘window’. (e) Each wavelength of light will generate its own window (three colours only shown here for clarity). (f) A rainbow hologram is made with each H1 window reduced to a narrow slit (three colours only shown here for clarity) it is possible to ‘see behind’ the image, as expected. The loss invertical parallax is due to the fact that the narrow slit excludes information upon this vertical direction from the holographic record. This is of little consequence when rainbow holograms are used for security-related purposes.

6.11.5 Hologram recording media

The interference pattern that is the information content of a hologram is stored in a photosensitive material. To some extent, the type of photoresponsive medium used depends on the purpose of the hologram. For artwork or security labels a thin hologram may sufﬁce; for data storage and retrieval a thick hologram, able to store multiple ‘sheets’ of data in the same volume may be necessary. The photosensitive response can involve a change of optical absorption, refractive index, optical anisotropy or thickness. The information can then be imprinted on the beam used for reconstruction by these changes. 241 Colour Due to Diffraction

(d) laser beam

H1 window

(e) white light red H1 window

violet H1 window

(f) white light

rainbow hologram Figure 6.29 (Continued)

For example, if the hologram is stored by a change in the optical absorption of the recording medium, the amplitude of the beam used for reconstruction is modulated. By analogy with diffraction grating nomenclature, this type of hologram is called an amplitude hologram. In cases where the refractive index or the thickness of the recording film is modulated, the information is impressed upon the reconstruction by changes in phase, and these holograms are called phase holograms. The use of optical anisotropy, particularly of refractive index, which will affect the polarisation of the reconstruction beam, gives rise to polarisation holograms. The first holographic recording medium used was fine-grain photographic emulsion deposited on a glass plate for maximum dimensional stability or else on a stable polymer base. Exposure to the object and reference beams causes silver crystallites to form in the emulsion (see Section 10.19). The emulsion, after development, contains blackened lines corresponding to the peaks in the interference pattern and transparent areas corresponding to the troughs. In this form the processed emulsion acts as an amplitude hologram. These primary holograms can be ‘bleached’ to remove the silver grains. After this process the refractive index of the emulsion varies in a mirror of the amplitude variation. In this form the emulsion acts as a phase hologram.It Colour and the Optical Properties of Materials 242 should be noted that commercially available photographic emulsions and films are becoming difficult to obtain (2010) and many hologram makers now prepare photographic emulsions themselves. Dichromated gelatine (DCG) deposited upon a glass or stable polymer film is a widely used phase hologram recording medium. The material consists of gelatine, a form of the natural product collagen, a widely distributed fibrous protein found in bones and connective tissues. The gelatine needs to be sensitised with 2 dichromate, Cr2O7 . On exposure to light, particularly violet or near ultraviolet, the dichromate causes the gelatine molecules to cross-link, giving the region a higher refractive index than the unexposed volumes. After appropriate treatment the interference pattern is fixed to give a phase hologram. This material is one of a family of photopolymers that record information in a similar way. The problems with this group centre upon long-term stability of the irradiated volumes or dimensional changes caused by shrinkage. There are a number of inorganic materials that are also being explored for recording holograms that use refractive index changes. Most studies have been carried out on ferroelectric crystals, such as lithium niobate LiNbO3. These are insulating oxides. Irradiation with light can excite electrons within the structure, usually from deliberately added dopant impurities such as Fe2 þ . (For details of these processes, see Chapters 7 and 10). Once the electrons have been formed they are free to move through the conduction band of the crystal until they come to another impurity atom or ion which is able to trap them. In effect, electrons shy away from the illuminated volumes and congregate in the dark regions. This charge distribution pattern not only mimics the irradiance distribution of the interference pattern, but also causes refractive index changes throughout the crystal volume. These record the information as a phase hologram. A number of inorganic glasses, especially of the chalogenides (S, Se and Te), are also utilised to record phase holograms. In these solids, light of sufficiently high irradiance is used to break some of the bonds in the glass. This leads to refractive index changes and, hence, to a phase hologram. Polarisation holograms are formed by embedding linear molecules with a dipole moment in a polymer matrix. A widely explored group of molecules considered for this application are derived from azobenzene modified by adding side chains to the benzene rings. When such a material is irradiated with linearly polarised light, the side chains orient parallel or perpendicular to the electric field vector of the light, causing the initially isotropic polymer matrix to become birefringent with an accompanying large change in refractive index. Irradiation with circularly polarised light can erase the hologram.

6.11.6 Embossed holograms

Embossed holograms are the physical holograms that are used as security labels. They are thin rainbow holograms made in the following way. The original master hologram H1 is a slit transmission hologram made as described above (Figure 6.30a). The H1 master is used to make a rainbow transmission hologram H2 which is made in a photoresist (Figure 6.30b). This is a polymeric material that reacts to light in one of two ways: the photoresist becomes insoluble (a negative photoresist) or soluble (a positive photoresist) on illumination with ultraviolet light. This photoresist layer is of the order of 1 mm thick and is mounted on a glass plate. The interference pattern making up the hologram causes the photoresist to react. Processing of the photoresist dissolves parts of the upper layers, which then leaves H2 as a grooved and ridged surface with either peaks or troughs corresponding to the peaks of the interference pattern (Figure 6.30c). The result is a phase hologram. This matrix is sprayed with a thin layer of silverpaint to make it electrically conducting and then nickel plated to give a thin nickel foil, the master or mother shim, which is the negative of the photoresist surface (Figure 6.30d and e). This is detached from the photoresist and used to make further copies, child or stamper shims, by further electroplating onto the prepared mother shim (Figure 6.30f). These are positive copies and are detatched from the mother shim for use. As many child shims as are needed are formed in this way. The child shims are used in conventional embossing machines (Figure 6.30g). These press the shim, under an appropriate 243 Colour Due to Diffraction

(a) object (b) slit reference beam

cylindrical lens hologram H1 H1 H2 on photoresist (c) (d) (e)

develop electroplate mother shim

(f) child shim (g) embossing shim

mother shim plastic film Figure 6.30 Embossed holograms (schematic): (a) preparation of slit hologram H1; (b) preparation of H2 on photoresist; (c) etched photoresist, forming a phase grating; (d) electroplate photoresist; (e) separate mother shim; (f) electroplate mother shim give a chid shim; (g) emboss plastic film with child shim temperature and pressure regime, into the surface of a thermoplastic sheet so that the surface relief is now imposed upon the polymer film. The film is backed with a reflective layer, which can be aluminium, zinc oxide, titanium dioxide or another plastic film, depending upon the ultimate use, and a layer of adhesive if needed. Naturally, the embossing machines operate on rolls of plastic and so can produce many thousands of holograms quickly. The initial preparation of the H1 slit master is costly and extremely difficult to duplicate. However, once in production the stamped copies cost just a few pence each. The combination of these two features, coupled with the unique appearance of these holograms, makes them ubiquitous both for security marking and for decorative purposes.

Further Reading

The classical theory of optical diffraction is described by E. Hecht, Optics, 4th edition, Addison-Wesley, San Francisco, CA, 2002. The prevention of diffraction spreading of a light beam is described by K. Dholakia, Nature 451, 413 (2008) and references cited therein. The way to obtain the wavelength of light using a steel rule or similar grating and some informative background to the method is given in W. P. Trower (ed.), Discovering Alvarez, University of Chicago Press, Chicago, IL, 1987, p. 1. An introduction to crystal structures and X-ray diffraction is R. J. D. Tilley, Crystals and Crystal Structures, John Wiley and Sons, Ltd, Chichester, 2006. The corona around the sun or moon and the patterns formed by droplets or dust on mirrors or other glass surfaces is described by Colour and the Optical Properties of Materials 244

J. Walker, Sci. Am. 245 (August), 116 120 (1981). D. K. Lynch, W. Livingston, Color in Light and Nature, Cambridge University Press, Cambridge, 1995, Chapter 4. C. F. Bohren, What Light Through Yonder Window Breaks? Dover, New York, 2006 (originally published by John Wiley and Sons, Inc., New York, 1991), Chapter 2. The microstructure of precious opal is described in J. V. Sanders, Acta Crystallogr. Sect.A24, 427 434 (1968). J. V. Sanders, P. J. Darragh, Mineral. Rec. 2 (6), (1971). P. J. Darragh, A. J. Gaskin, J. V. Sanders, Sci. Am. 238 (April), 87 95 (1978). There are currently large numbers of research papers concerned with inverse opals, colloidal crystals and photonic crystals. Some starting points for further study are G. I. N. Waterhouse, J. B. Metson, H. Idriss, D. Sun-Waterhouse, Chem. Mater. 20, 1183 1190 (2008). S. John, Nature 460, 337 (2009). K. Ishizaki, S. Noda, Nature 460, 367 370 (2009). A. S. Iyer, L. A. Lyon, Angew. Chem. Int. Ed. 48, 4562 4566 (2009). For further information on photonic crystals and related materials, especially those that generate beautiful natural colours, see A. van Blaaderen, Mater. Res. Soc. Bull. 23 (October), 36 43 (1998). A. Parker, Proc. R. Soc. Lond. Ser.B262, 349 355 (1995). A. Parker, R. C. McPhedran, D. R. McKenzie, L. C. Botten, N.-A. P. Nicorovici, Nature 409, 36 37 (2001). P. Vukusic, Structural colour, in Dekker Encyclopedia of Nanoscience and Technology, Volume 5, J. A. Schwarz, C. I. Contescu, K. Putyera (eds), Marcel Dekker, New York, 2004, pp. 3713 3722. P. Vukusic, J. R. Sambles, Nature 424, 852 855 (2003). Various authors, Mater. Res. Soc. Bull. 26, 608 646 (2001). The topic of colour in nature is described from an evolutionary perspective, with examples of diffraction colours, by A. R. Parker, In the Blink of an Eye, Free Press, London, 2003.

For further information on diffraction by moth-eye structures and high spatial-frequency surface gratings, see T. K. Gaylord, W. E. Baird, M. G. Moharam, Appl. Opt. 25, 4562 4567 (1986). T. K. Gaylord, E. N. Glytsis, M. G. Moharam, Appl. Opt. 26, 3123 3135 (1987). A. R. Parker, Am. Sci. 87, 248 255 (1999). Y. Ono, Y. Kimura, Y. Ohta, N. Nishida, Appl. Opt. 26, 1142 1146 (1987). A very interesting description of an AR surface grating on the eye of a 45 million-year-old ﬂy preserved in amber is given by A. R. Parker, Z. Hegedus, R. A. Watts, Proc. R. Soc. Lond. Ser.B265, 811 815 (1998). The cholesteric colours of certain beetles and how these may be proved to come from twisted layered structures is given by A. C. Neville, S. Caveney, Biol. Rev. 44, 531 562 (1969). The cholesteric blue phases are described by P. H. Keyes, Mater. Res. Soc. Bull. 16 (January), 32 37 (1991). 245 Colour Due to Diffraction

The structure of the cornea is given by J. P. Giraud, Y. Pouliquen, G. Offret, P. Payrau, Exp. Eye Res. 21, 221 229 (1975). The structure and colour of the Plum-throated Cotinga is described by R. O. Prum, R. H. Torres, S, Williamson, J, Dyke, Nature 396, 28 29 (1998). Simple descriptions of making holograms, interesting from a historical point of view, are J. Walker, Sci. Am. 242 (February), 124 128 (1980). J. Walker, Sci. Am. 260 (May), 100 103 (1989). See also F. Unterseher, J. Hansen, B. Schlesinger, Holography Handbook, Ross Books, 1996. P. Hariharan, Basics of Holography, Cambridge University Press, 2002. An extremely clear explanation of rainbow holograms is found in the tutorial by K. Bazargan, at http://holographer.org (2004).

7 Colour from Atoms and Ions

. How can the chemical composition of the sun and other stars be determined? . How do sodium street lamps produce yellow light? . Why are transition metal compounds coloured?

In the preceding chapters, colour generation has been described in terms of the wave theory of light. This no longer sufﬁces, and in much of the remaining material, starting with this chapter, photons and quantum ideas are necessary.

7.1 The Spectra of Atoms and Ions

The spectrum of electromagnetic radiation emitted by an incandescent solid (Sections 1.6 and 1.7) is continuous and contains all wavelengths. The spectrum of light emitted by a rareﬁed gas of atoms or ions consists of a sequence of bright lines. For example, if the light from mercury or sodium street lamps is dispersed by an inexpensive diffraction grating positioned in front of a camera lens, the image will consist of a set of individual coloured copies of the light source, each of which arises from a particular line in the spectrum (Figure 7.1a). Narrowing the source image to a slit, as in a spectrometer,1 will yield a series of sharp lines when an excited gas is the source (Figure 7.1b). This latter pattern is called a line spectrum. If a continuous spectrum

1 Devices for the display of spectra are called spectroscopes, spectrographs or spectrometers. They use prisms or diffraction gratings and the resultant spectra are recorded and displayed electronically.

Colour and the Optical Properties of Materials Richard J. D. Tilley 2011 John Wiley & Sons, Ltd Colour and the Optical Properties of Materials 248

(a)

(b)

Wavelength / nm 690.8 577.0 546.1 433.6 404.7 579.1 434.7 435.8 Figure 7.1 (a) The image of a mercury-vapour street light taken with an inexpensive transmission diffraction grating over the lens. Each coloured lamp image arises from one line in the ﬁrst-order diffraction spectrum of mercury. The white lamp at the right is the zero-order image and the deep blue image at the far left is the beginning of the second-order series. (b) The line spectrum of mercury vapour. The bright lines correspond to the images in (a) as shown

from an incandescent source is passed through a gas of the same atoms then the bright line spectrum will be shown as a negative and appear as a set of dark lines on the continuous bright background (Section 7.4). In this chapter, the origin of these lines is described. For most chemical purposes an atom or an ion can be considered to consist of a dense minute nucleus surrounded by electrons which are said to occupy a series of orbitals. The electron configuration of an atom or an ion describes theway in which these electrons are allocated to these orbitals (Appendix A7.1 and Figure 7.2). The spectral lines emitted or absorbed (the source of colour of atoms) arise when electrons jump from one orbital to another. The energies of the orbitals in isolated atoms and ions are precise and the total energy of all of the electrons is then represented as a sharp energy level. Energy is absorbed when electrons are excited from a lower energy level to a higher level and exactly the same energy is released when the electron drops back to the same lower level again. From this point of view, the spectral lines should be infinitely narrow, but because of atom motion the lines are broadened due to the Doppler effect. The broadening amounts to v /c, where v is the velocity of the atoms and c the velocity of light. Doppler broadening is thus greatest for light atoms at high temperatures. Other interactions, notably electron electron and electron nucleus, which are described by quantum electrodynamics, also place limits upon the sharpness of the lines, and even in ideal circumstances the lines have a width termed the natural linewidth. These refinements explain the shape of spectral lines, but the main point remains that the energy of the line is centred upon the energy separating the final and initial states that the electron occupies. When the energy of the radiation absorbed or emitted falls in the visible these transitions give rise to colours. For transitions giving 249 Colour from Atoms and Ions

Group 1 2 3456789 10 11 12 13 14 15 16 17 18 IA IIA IIIB IVB VB VIB VIIB VIIIB IB IIB IIIA IVA VA VIA VIIA VIII Period 1

Figure 7.2 The Periodic Table of the elements, giving the outer electron conﬁguration (below element symbol) and ground-state level (upper right) of the atoms. The transition metals are coloured orange, the lanthanoids blue and the actinoids mauve. Note that the terms "lanthanoid" and "actinoid" are now preferred by IUPAC to "lanthanide" and "actinide" (see footnote to Appendix A7.1.3) rise to lines in the visible, 400 700 nm, the width of a spectral line at ordinary temperatures is about 0.0005 nm. A light photon can only interact with an electron if it has the exact amount of energy to allow an electron to pass from one precise energy level to another. Thus, when a photon of energy hn is absorbed by an atom or ion it passes from a lower energy state, often the lowest available state in the system, called the ground state E0,toanupper one E1 (Figure 7.3a). The transition will only take place if the frequency n of the photon is given exactly by:

D n ¼ E1 E0 ¼ E h h where DE is the energy separation of the two energy levels and h is Planck’s constant. If the atom is in the upper state E1 and makes a transition to the lower state E0, the same quantity of energy, DE, will be emitted (Figure 7.3b). This will have the same frequency, given by the same equation:

D n ¼ E1 E0 ¼ E h h

Each transition gives rise to a line in the spectrum (Figure 7.3c). The line spectra of most atoms are complex and contain large numbers of lines, but hydrogen has a relatively simple visible spectrum, which contains four strong lines (Figure 7.4a) and because of this has played a Colour and the Optical Properties of Materials 250

(a) E1 (b) E1

hν hν

E0 E0 light absorbed light emitted

(c) Intensity ν Frequency Figure 7.3 The absorption and emission of radiation by isolated atoms or ions. (a) Light is absorbed if the energy of the photon exactly matches the energy gap between the lower level and the upper level. (b) When energy is released spontaneously the energy of the photon is again exactly equal to the difference in energy between the upper and lower levels. (c) The emission from a collection of atoms in a gas at low pressure will consist of a narrow line. The frequency at which the line occurs is the same as that of the photons involved in absorption and emission in (a) and (b) prominent role in twentieth-century physics. The lines form part of a series, the Balmer series, with positions given by the formula: 1 1 1 n ¼ ¼ RH l 22 n2

1 where n is the wavenumber (usually measured in cm ; Appendix A1.1), RH is the Rydberg constant (1.097 107 m 1 ¼ 1.097 105 cm 1) and n takes values of 3, 4, 5, ...). As the formula shows, the lines 2 gradually approach each other and reach a series limit as n approaches inﬁnity, given by RH/2 (Table 7.1). This series was explained theoretically by Bohr in his celebrated theory of the hydrogen atom. The lines are caused by transitions of an excited electron from the n ¼ 3, 4, 5, ...shells to the n ¼ 2 shell of the atom (Figure 7.4b). Because the ns, np, nd, etc. orbitals of hydrogen all have the same energy, it does not matter which exact orbital the electron is excited to or falls from, and this is the reason for the apparent simplicity of the spectrum. (In fact, the lines all are multiplets and the explanation of these features required the profound

Table 7.1 The Balmer series of visible spectral lines of atomic hydrogen

n Designation l/nm l 1/cm 1 3Ha 656.3 15 237 4Hb 486.1 20 572 5Hg 434.0 23 041 6Hd 410.2 24 378 ¥ 364.7 27 420

Spectral lines are often speciﬁed in terms of n ¼ 1=l given in units of cm 1 (see Appendix A1.1). 251 Colour from Atoms and Ions

(a)

410 2 434.1 486.1 656 3

(b) 13.6 eV n = ∞ n = 4 n = 3 656.3 nm 486.1 nm 434.0 nm 410.2 nm

H-α H-β H-γ H-δ n = 2

Balmer series Energy

0 n = 1 Figure 7.4 (a) The visible spectrum of atomic hydrogen. The wavelength of each line is noted in nanometres. At high resolution these lines are seen to split into more complex groups of lines. (b) The transitions to the n ¼ 2 level of the hydrogen atom give rise to the visible spectral lines in the Balmer series development of quantum theory; see this chapter’s Further Reading.) Other series form as electrons fall from excited levels to the n ¼ 1 (Lyman), n ¼ 3 (Paschen), n ¼ 4 (Bracket), n ¼ 5 (Pfund) orbitals, but these lie outside the visible. The spectra of atoms containing many electrons, even of ‘one-electron’ atoms such as lithium and sodium, are far more complex, and the explanation of these spectra was one of the great triumphs of twentieth-century science. Each spectral line gives precise information about the difference in energy between the two energy levels involved in the electron transition, and this can be indicated by using a formula for the frequency or wavenumber containing the difference between these two quantities. Thus, the Balmer series can be written in the form:

n ¼ 1 ¼ l T1 T

2 Tl and T are called terms,withTl representing the series limit RH/2 . All other spectral lines can be described in a similar way, as the difference between two terms. How these terms are derived follows. Colour and the Optical Properties of Materials 252

7.2 Terms and Levels

The electron configurations of atoms or ions (Appendix A7.1, Figure 7.2) are a first-level approximation as far as energies are concerned. They do not even take electron electron repulsion into account and are worked out by assuming that there is only one electron circling a nucleus surrounded by a negative cloud made up of all of the other electrons present. Because only one electron is involved in the computations the quantum numbers are called hydrogen-like or one-electron symbols and they are given lower case letter labels. Atomic or ionic spectra, on the other hand, consist of a series of lines which give information on the exact energy difference between two energy levels in the species under investigation. A measurement of atomic spectra thus allows the real energy levels of atoms to be assessed. The energy levels of an isolated atom aregiven labels called term symbols.Aterm is a set of states or energy levels which arevery similar in energy. Transitions between these terms, or, more precisely, the energy levels that make up the set of states specified by the term, give rise to the observed line spectrum of an atom. Term symbols are derived by taking into account electron interactions. The most important of these, for an understanding of spectra, are electron electron repulsion and the combining, or coupling, of the orbital and spin angular momenta. There are a number of ways of carrying out this coupling. The best known method, which is mainly applicable to light atoms, is called Russell Saunders or LS coupling (Appendix A7.2). In this designation, each term is written as 2S þ 1L, where L is a many-electron quantum number describing the total orbital angular momentum of all of the electrons surrounding the atomic nucleus and S is a many-electron quantum number representing the total electron spin. Sometimes the term symbol has an initial value n, when all of the electrons outside the core (that is, those involved in transitions) come from the same shell. The superscript 2S þ 1 is called the multiplicity of the term. Upper case letters are used to make it clear that all electrons are included and to differentiate them from the hydrogen-like configurations. Terms, therefore, apply to the overall energy state of the atom or ion as a whole. The total angular momentum quantum number L is replaced by a letter symbol similar to that used for the single electron quantum number l. The correspondence is set out in Table 7.2. After L ¼ 3, F, the sequence of letters is alphabetic, omitting J. Be aware that the symbol S (italic) means the value of total spin while S (roman) gives the value of L. As an example, the alkali metal atoms in their lowest energy state all have a single ns electron outside of a closed shell. All core electrons can be ignored, so that the only electron to consider when constructing the term is the outer s electron. Thevalue of L must be equal to thevalue of l for the electron, i.e. zero, so that the state is S. 1 ¼ The spin on the electron is 2, so that the total spin quantum number S (do not confusewith that just used for L 0) 1 þ ¼ 2 is 2. The multiplicity is then (2S 1) 2. The lowest energy term for all of the alkali metals is thus S. Appendix A7.2 sets out in detail the way in which the terms of an atom can be derived.

Table 7.2 The correspondence of L values and letter symbols

L Symbol 0S 1P 2D 3F 4G 5H 253 Colour from Atoms and Ions

Even the term symbol does not account for the true complexity found in most atoms. This arises from the interaction between the spin and the orbital momentum (spin orbit coupling) that is ignored in Russell Saunders coupling. For this the quantum number J is needed. It is given by:

J ¼ðL þ SÞ; ðL þ S1Þ; ...; jLSj where |L S| is the modulus (absolute value, irrespective of whether þ or ) of the quantity L S. The new 2S þ 1 quantum number is incorporated as a subscript to the term, now written LJ and this is no longer called a term, but a level. Each value of J represents a different energy level. Thus, the term for the alkali metals 2S can be þ ¼ 1 expanded by noting that the value of the quantum number J is given by L S 2. The ground-state energy level 2 term for all of the alkali metals is thus S1=2. To specify the ground state of a sodium atom, for example, the fact 2 that the outer electron is in the 3s orbital would allow the level to be written 3 S1=2. It is found that a singlet term always gives rise to one level, a doublet to two, a triplet to three and so on. The progression from the electron configuration of an atom to a set of energy levels thus involves a number of steps, shown schematically for a 3d2 ion such as Ti2 þ or V3 þ in Figure 7.5. At the far left of the figure, the electron configuration is shown and it is assumed that the ion can be represented by a single energy level. This is useful chemically, but is unable to account for the spectra of the atom. Russell Saunders coupling is a reasonable approximation to use for the 3d-metals, and the terms that arise from this are given to the right of the configuration. In Russell Saunders coupling the electron electron repulsion is considered to dominate the interactions. The ion is now allocated five energy levels, the lowest being represented by a term 3F. The terms are split further if spin orbit coupling (j j coupling) is introduced. The number of levels each term forms is the same as the multiplicity of the term, 2S þ 1, and this leads to nine energy levels in total. The procedure for dealing with atoms in general is given in Appendix A7.2 and the ground-state level of all atoms can be found in Figure 7.2.

1 S 1 S0 1 G 1 G4

3 P2 3P 3 1 P 3 P0

1 D 1 D2

3 F4

3 F3

2 3 3d F 3 F2 No Electron–electron Spin–orbit interaction repulsion (Russell- interaction (one-electron Saunders terms) (j–j levels) configuration) Orbitals Terms Levels þ þ Figure 7.5 The schematic development of the energy levels of a free d2 ion, such as Ti2 or V3 Colour and the Optical Properties of Materials 254

In a heavy atom it might be preferable to derive the energy levels by proceeding from the electron configuration to levels derived by j j coupling and then add on a smaller effect due to electron electron repulsion. In real atoms, the energy levels determined experimentally are often best described by an intermediate model between the two extremes of Russell Saunders and j j coupling, and for these atoms alternativecoupling schemes may be preferred. The splitting of terms into levelsdue to spin orbit coupling is of considerable importance for the lanthanoids. In addition, note that in the presence of a magnetic field these spin orbit levels are split further, so that the spectra of atoms and ions in magnetic fields are more complex than that already discussed. This has relevance not only for laboratory work, but also for the interpretation of stellar spectra, where extremely strong magnetic fields can occur. The same is true of static electric fields. In both cases, atoms or ions in a gas or free space will show an average effect because of the motion of the particles. However, in a crystal, the atom and ion positions are more or less fixed and the application of either magnetic or electric fields along certain symmetry directions will, in general, cause different degrees of splitting of the levels than the same fields applied along different symmetry directions.

7.3 Atomic Spectra and Chemical Analysis

Although the terms and levels give a picture of the energy levels available within an atom, the spectra cannot be explained simply by working out all of the possible transitions between them. Apart from the energy restriction mentioned above, light can only interact with electrons if the wave functions specifying the initial and final states fulfil certain conditions. This latter restriction leads to a number of selection rules which allow one to determine whether the transition is probable or improbable. These selection rules depend upon how the light interacts with the electrons. The quantum mechanical interactions that the wave functions describe can be specified in terms of an interaction of the electric field of the light wave with the electrons, a situation known as an electric dipole transition, or with the magnetic field of the light wave, resulting in a magnetic dipole transition. (Other less frequently observed transitions involving electric and magnetic quadrupoles and more complex configurations are also possible but become increasingly rare.) The selection rules which set out which transitions are allowed and which are not allowed actually represent the probabilities of the transition occurring. Thus, allowed transitions have a very high probability of taking place, whereas forbidden transitions have a very low probability of occurring, but are not absolutely forbidden. Electric dipole transitions have the highest probability, giving rise to intense lines in the spectrum of an atom and so are of primary importance in colour production. The other types of transition have lower probabilities of occurring, and at best give rise to weak lines in the spectrum. The selection rules applicable to Russell Saunders terms and levels leading to the absorption or emission of light due to electric-dipole transitions are n no restriction DS cannot change DL can change by 0, 1, but L ¼ 0toL ¼ 0 is forbidden DJ can change by 0, 1, but J ¼ 0toJ ¼ 0 is forbidden.

In addition, the Laporte selection rule, sometimes called the parity selection rule, limits transitions only to those for which the symmetry of the wavefunctions specifying the start and end states are of opposite parity. It means that transitions between orbitals of the type s to s, p to p, d to d and f to f are all forbidden. This has considerable importance for the operation of lasers, as forbidden transitions (i.e. transitions with a low probability) are 255 Colour from Atoms and Ions

Table 7.3 Flame test colours

Atom Colour Atom Colour Atom Colour lithium scarlet sodium yellow calcium orange red copper blue potassium violet strontium crimson rubidium red violet barium green caesium blue associated with energy states with long lifetimes. These are necessary to obtain the population inversions needed for laser action (Section 1.9). The exact arrangement of the energy levels in an atom is very sensitive to the electron configuration. When this constraint is coupled to the selection rules operating, it emerges that the line spectrum of each chemical element is unique. Thus, the spectrum becomes a powerful analytical tool. Each atom or ion can be thought of as having a line spectrum fingerprint which can be used as a diagnostic test for the element. At the simplest level this is made use of in inorganic chemistry as a ‘flame test’. A small quantity of the materialbeing examined is placed upon a platinum wire and heated to high temperature in a flame. The colour of the flame is a guide to the atoms present. This method works well with the alkali metals and alkaline earth metals, which produce clearly identifiable colours (Table 7.3). At this point it is well to be aware that the colours produced in a flame, or in fireworks, which are similar, are the result of complex interactions and frequently arise from molecular species rather than isolated atoms or ions. Thus, the scarlet colour arising from lithium compounds is due to radiation from LiOH molecules rather than isolated Li or Li þ . Green colours from barium compounds are derived from the molecular species BaCl þ and BaOH þ , and the red of strontium compounds derives from SrOH þ and SrCl þ rather than Sr or Sr2 þ . The same could be said about other flame and firework colours. Colour from molecules is described in Chapter 8. Much more information about the colours given out by the flame can be obtained by allowing the light to pass through a narrow slit and viewing it with an inexpensive plastic diffraction grating. The grating spreads the light out into a series of spectra which in the case of atoms or ions consist mainly of lines and which in the case of molecules also contains bands (see Chapter 8). In fact, such an arrangement is a simple spectroscope. The technique can yield more information if the intensities and positions of the lines in the spectrum can be recorded, and it is this technique which allows one to determine that molecular species are important in flame colours rather than isolated atoms or ions. Comparison of the intensities of lines with those from standard solutions of ions allows quantitative analysis of even very small quantities of impurities to be made. The technique is called atomic absorption analysis. It is routinely used to detect quantities of metal impurities at concentrations of parts per million.

7.4 Fraunhofer Lines and Stellar Spectra

In 1814, Fraunhofer, by making better spectrographs than any others available at that time, discovered that the solar spectrum was interspersed with a number of dark lines, now called Fraunhofer lines. The most important visible Fraunhofer lines are illustrated in Figure 7.6 and listed in Table 7.4. These features are actually absorption spectra and consist of both sharp lines and wider bands. The lines are due to absorption by single isolated atoms and ions, whereas the bands arise from molecules that lie between the source of light and the observer. (The reasons why molecules often give rise to absorption bands is described in Chapter 8.) Colour and the Optical Properties of Materials 256

α G FE2 D3 D2 D1 C B

400 500 600 700 Wavelength / nm Figure 7.6 The main visible Fraunhofer lines and bands, visible as dark lines within the continuous spectrum of the sun. The atoms or molecules responsible for these features are listed in Table 7.4

Table 7.4 Some Fraunhofer lines

Designation Origin Wavelength/nm

BO2 molecules 687.7 688.4 (band) C hydrogen, H a 656.2 a a, O2 molecules 627.6 628.7 (band) D1 sodium 589.6 D2 sodium 589.0 D3 or d helium 587.6 E2 iron 527.0 F hydrogen, H b 486.1 G0 hydrogen, H g 434.0 G iron, calcium 430.8 h hydrogen, H d 410.2

The Fraunhofer lines arise in two ways. One set of absorption lines and bands, the telluric lines, are due to components of the Earth’s atmosphere. These absorb incoming solar radiation and give rise to dark lines or bandsintheotherwisecontinuousspectrumfromthesun.Theprincipalcontributionsarefromoxygenandwater vapour. However, another set of lines arises when light from the sun is absorbed by atoms or ions in the relatively cool outer solarregions.Among the most prominent ofthese are lines from hydrogen, sodium, calcium and iron, which could all be identified by comparison with spectra from standards available in the laboratory. Significant among the information, a Fraunhofer line at a wavelength 587.6 nm, discovered in 1868, could not be attributed to any known element. The new line was taken as an indication of the presence of a new element in the solar atmosphere. The element was subsequently named helium, from the Greek word for the sun, helios. Almost 30 years were to pass before the gas was discovered on Earth, by Ramsay, who isolated it in 1895. Nowadays, the presence of metallic atoms and ions in the outer atmospheres of stars or even far-off galaxies is generally confirmed by recording the spectrum of the star and examining the dark absorption Fraunhofer lines found.

7.5 Neon Signs and Early Plasma Displays

Faraday, in 1835, ﬁrst discovered that gases at low pressure could conduct electricity and at the same time give out light. The complex processes taking place were investigated in depth by Geissler in the 1860s and at the end of the 1890s by Crookes. 257 Colour from Atoms and Ions

The experimental observations are easy to report. If a gas is contained in a tube and is at atmospheric pressure it will not conduct electricity unless it is subjected to an extremely high voltage, as when lightning strikes through the atmosphere. If the pressure is reduced and the gas is subjected to a voltage of the order of kilovolts it begins to show electrical conductivity and at the same time it starts to emit light. The colour of the light depends upon the gas in the tube. This is the basis for the operation of neon signs, invented in 1910, and of sodium- and mercury-vapour street lighting. Ultimately, as the pressure falls, the number of spectral lines diminishes, until only the so-called ‘persistent lines’ can be observed. Finally, all light emission ceases and the tube no longer glows. During the first half of the twentieth century the gas pressure in scientific vacuum equipment, including electron microscopes, was estimated by using the light emitted when an electric field was imposed on the residual gases in a tube (a Geissler tube), which connected with the main body of the equipment. When the glow had been totally extinguished the vacuum was usually as good as could be obtained with the then available vacuum pumps and was colloquially called a ‘black vacuum’. In Geissler tubes, neon signs and similar devices, electrons are emitted from the cathode (the negative electrode) and are accelerated in a high electric field across the gas. These electrons collide with gas molecules and excite them to higher energies. Some molecules are ionized and these ions in turn are accelerated in the electric field and cause further ionization and excitation. The light emitted is due to the excited atoms and ions losing energy by releasing photons as they return to lower energy states. Lamps that emit light by this mechanism are generally termed gas discharge lamps. ‘Neon’ signs are gas discharge lamps that make use of atoms of the inert gases as the working medium. These elements exist as monatomic gases at normal temperatures and all can be used in what are now collectively known as neon signs, neon being the first to be used. To make a neon sign, a glass tube is evacuated and filled with a low pressure of one of the inert gases. The gas is subjected to a voltage of about 10 kV via electrodes at opposite ends of the tube, which causes it to glow with a characteristic colour (Table 7.5). In some displays a mixture of gases is used to change the apparent colour of the glow. Gas discharge tubes using xenon (Xe) as the working medium are also widespread. Under low pressures and fairly low current densities, the light takes on a blue hue and is high in ultraviolet radiation. However, when the gas is present at rather higher pressures, and when the current density becomes high, the tubes emit lightthatisperceivedtobewhite.Thisismadeuseofin two common ways. Xenon arc lamps, operating at high pressures (up to 300 atm) and temperatures give out continuous light. They are notably used in cinema projectors. Flashlamps or flashtubes are used to give a high-intensity white-light output over an extremely short interval. These contain low pressures of xenon (up to 0.1 atm). The flash is achieved by sending a large pulse of current through the gas via a charged capacitor. Xenon flashtubes are commonly found as the flash unit in cameras.

Table 7.5 Colours produced by the inert gases

Gas Colour Helium yellow Neon pink red Neon þ argon red Argon pale blue Argon þ mercury blue Krypton lavender Xenon blue Colour and the Optical Properties of Materials 258

The energy level diagrams of the inert gases are complex because the simplest excited state, [np5 (n þ 1)s], consists of two unfilled shells. Each configuration of the excited atom thus gives rise to a considerable number oflevels.These are mostoften displayed inthe form of a Grotrian diagram, in which the terms are set out as a set of ladders of increasing energy. The terms with a given multiplicity all form a single ladder. Spectral transitions are indicated by lines between relevant terms: absorption represented by an upward transition and emission by a downward transition. A simplified version of such a diagram for neon, which shows electron configurations rather than terms (Figure 7.7), reveals that the main lines contributing to the red colour are due to transitions from the excited 2s2 2p5 3p configuration, comprising ten levels in total, to the 2s2 2p5 3s excited state. These include the two most intense lines, at 692.9 nm and 703.2 nm. The ground state, with configuration 2s2 2p6 with 1 a single level, S0, does not figure in transitions giving rise to visible radiation. In the case of xenon, the spectrum consists of relatively few lines at low excitations, giving rise to the blue tint of the output light. However, there are so many transitions possible that can produce light when the excitation energy is high that the output appears to be white, although the emission is still in the form of lines and is not a continuous spectrum. The observed colour changes from blue to white as the discharge energy increases. Similar colour changes occur with the other gases mentioned.

(a)

(b) 22

6d 6s 7s 5d 21 170000 5p 4d 5s 478 9 471.5 4p 3d 20 4s 160000 471 0, 470.4, 453.8

488 5, 482.7 Energy / cm 19 576.4, 534.1 3p 150000 –1

Energy / eV 18

703 2, 692.9, 650.6, 640 2 140000 17 3s

16 130000

2s22p5 ns: 2s22p5 np: 2s22p5 nd: 4 levels in 10 levels in 12 levels in 15 each box each box each box Figure 7.7 (a) Line spectrum of neon. (b) Schematic Grotrian energy-level diagram for neon with conﬁgura- tions rather than terms displayed. The ground state, at energy zero, is not shown. Each excited state gives rise to a number of levels which are not drawn individually but represented schematically by a box. The transitions giving rise to emission colours are drawn as downward-pointing arrows. The wavelengths marked (in nanometres) are just a few of the more intense lines in the visible. (Conversion of the units eV and cm 1 to the SI unit joule (J) is given in Appendix A1.1) 259 Colour from Atoms and Ions

The state of matter giving rise to the colours is a form of plasma, in which the high electrical field imposed across the tube strips the enclosed atoms of some electrons and creates a fluid consisting of positive and negative entities. The plasma usually forms close to the cathode and is localized in a small volume giving out high-intensity light. In some lamp designs, plasmas can form at both electrodes, thus increasing the output. Similar plasma colours can occur in the atmosphere, although potentials of the order of 30 000 V cm 1 are needed. The best known natural display of this type is St Elmo’s fire. This a deep blue or violet glow appearing around tall and generally sharp objects, especially, in historic times, the masts of sailing ships.2 It occurs sporadically and is particularly observed when the weather is heavy and thundery. It is believed that high static electric fields, enhanced in the neighbourhood of sharp points, are intense enough to break apart the molecules in the surrounding air, mostly nitrogen and oxygen, to form a local plasma of ionised fragments. The release of energy as the excited fragments regain the ground state gives rise to colours in an analogous way to those in neon lights. Note that, as in the case of flame tests described above, the actual constituents involved, which are likely to contain molecular fragments as well as ionized single atoms, are not well understood. With the advent of portable computers a need arose for a lightweight flat display screen. Among the first of these was the monochrome gas plasma display, which, having been developed some years previously, operated on the principles just outlined. Ionised inert gases (mainly neon) were employed to produce the illumination. A photograph of a display of this type, from a computer available in 1989, is shown in Figure 7.8. This technology rapidly gave way to full-colour plasma displays, which are described in Section 9.5.

7.6 The Helium–Neon Laser

For laser action, two objectives have to be fulfilled (Section 1.9). It is necessary to obtain a population inversion between two energy levels and then ensure that the higher energy level is depopulated by stimulated emission, not by spontaneous emission. This was first achieved in the ruby laser by Maiman in 1960 (Section 7.11). However, not long after this, at the end of 1960, Javan constructed the helium neon (He Ne) laser. This laser is the ubiquitous red laser common in supermarket check-out counters and laser pointers, although colours other than red can also be produced by the helium neon combination; in fact, the first laser wavelength produced was at 1.15 mm. The laser consists of a low pressure (10 2 to 10 3 atm) of helium mixed with about 10 % neon, enclosed in a narrow glass tube. The laser uses excited helium to transfer energy to neon and so obtain a population inversion. The helium is excited by a high voltage, just as in a neon sign. High-energy electrons are produced by subjecting the cathode (the negative terminal) to a high voltage. These energetic electrons, e, collidewith the helium atoms to produce an excited state, He:

He ð1s2Þþe* ! He* ð1s1 2s1Þþe

2 A good description of St Elmo’s fire is given by Darwin at the start of Chapter III in his Journal of Researches ..., better known as the Voyage of the Beagle. The incident described takes place in the estuary of the River Plate. ‘On a second night we witnessed a splendid scene of natural fireworks; the mast head and yard arm ends shone with St. Elmo’s light; and the form of the vane could almost be traced, as if it had been rubbed with phosphorus. The sea was so highly luminous, that the tracks of the penguins were marked by fiery wakes, and lastly the sky was momentarily illuminated by the most vivid lightning.’ Later in the same chapter Darwin remarks that ‘the neighbourhood of the Rio Plata seems particularly subject to electric phenomena’. St Elmo was the patron saint of sailors, and the presence of St Elmo’s fire was believed to give protection against storms. Colour and the Optical Properties of Materials 260

Figure 7.8 A gas plasma ﬂat-screen display on a portable computer of 1989

1 1 1 3 The 1s 2s conﬁguration gives rise to two energy levels 2 S0 and 2 S1 (Figure7.9).TheexcitedHe can pass its energy over to a neon atom during a collision to produce an excited neon atom, Ne. This can happen because, quite by chance, the energy to be transferred is almost exactly the same as two excitation energy transitions of Ne:

ð 1 1; 1 Þþ ð 2 6Þ! þ ð 2 5 1Þ He* 1s 2s 2 S0 Ne 2s 2p He Ne* 2s 2p 5s

ð 1 1; 3 Þþ ð 2 6Þ! þ ð 2 5 1Þ He* 1s 2s 2 S1 Ne 2s 2p He Ne* 2s 2p 4s The neon energy levels derived from these two configurations, each consist offour energy levels. In addition, there are two other sets of 10 energy levels present on the neon atoms, derived from the configurations 2s2 2p5 3p1 and 2s2 2p5 4p1. (All of the energy levels on the neon atoms are complex and laser workers use a labelling system of s and p designations (Table 7.6). Unfortunately, this mimics the chemical configuration symbols, but without the same implications, which leads to unnecessary confusion. In order to relate Figure 7.9 with the labels found in laser texts, the correspondence in nomenclature is given in Table 7.6.) The collisions between neon and excited helium atoms (He) produces a population of excited neon atoms (Ne) in which several series of occupied and empty energy levels exist in close conjunction. These excited Ne atoms can release energy by stimulated emission, thereby dropping to many of the empty levels, and about 100 or more output wavelengths can appear. The main transition, however, is from the 2s2 2p5 5s1 set of levels to the 2s2 2p5 3p1 set of levels:

Ne* ð2s2 2p5 5s1Þ!Ne* ð2s2 2p5 3p1Þþhn 261 Colour from Atoms and Ions

22 He Ne 6s 170000 21 5p 1 5s 3391 nm 2 S0 1s1 2s1 4p 20 23S 4s 160000 1 633-543 Energy / cm

19 1152 nm 3p 150000 –1

Energy / eV 18 pump rapid decay 140000 17 3s

16 130000 wall collisions

2s2 2p5 ns: 2s2 2p5 np: 4 levels in 10 levels in each box each box Figure 7.9 Schematic processes operating in a helium–neon laser. Helium (He) atoms are excited from the 1s2 ground state into two 1s1 2s1 (He) states and subsequently transfer energy to neon (Ne) atoms to excite them to the 2s2 2p5 4s and 2s2 2p5 5s groups of energy levels. The main laser transition is from the 2s2 2p5 5s manifold to the 2s2 2p5 3p group of levels, from which the Ne atoms return to the ground state in two steps

The transition produces the well-known red laser output with a wavelength of 632.8 nm. Transitions to some of the other levels in the same manifold give rise to the other coloured output frequencies, which include 543.5 nm (green), 594.1 nm (yellow) and 612.0 nm (orange). The still-energized neon atom thereafter rapidly decays to the ground state, 2s2 2p6, in two steps:

Ne* ð2s2 2p5 3p1Þ!Ne* ð2s2 2p5 3s1Þ!Ne ð2s2 2p6Þ

The 2s2 2p5 3p1 to 2s2 2p5 3s1 transition is fast and helps to maintain a population inversion between the 2s2 2p5 3p1 level and those above it. The ﬁnal transition is radiationless and energy is often lost to the walls of

Table 7.6 Energy levels in neon

Electron conﬁguration Laser terminology 2 5 1 2s 2p 3s 1s2 1s5 (4 energy levels) 2 5 1 2s 2p 3p 2p1 2p10 (10 energy levels) 2 5 1 2s 2p 4s 2s2 2s5 (4 energy levels) 2 5 1 2s 2p 4p 3p1 3p10 (10 energy levels) 2 5 1 2s 2p 5s 3s2 3s5 (4 energy levels) Colour and the Optical Properties of Materials 262 the laser tube in transitions which do not give out light. The renewed population of ground-state He and Ne atoms allows the process to begin all over again.

7.7 Sodium and Mercury Street Lights

Sodium street lights give out a characteristic yellow colour, which arises from excited Na atoms and Na þ ions. As sodium is a solid at normal temperatures, the initial discharge is through a low pressure of neon which is also contained in the lamp tube. This ‘neon lamp’ is ﬁrst activated, which is the reason why sodium lamps glow with a pink red colour when they are warming up. After a short time the energy supplied to the neon generates enough heat for the sodium to evaporate. At this stage, collisions between electrons accelerated by the electric ﬁeld and sodium atoms excite these latter to higher energy levels. On falling back to the ground state this energy is released. A partial energy level Grotrian diagram of sodium atoms is given in Figure 7.10. This shows that, for neutral 2 2 sodium atoms, each return path traverses the closely spaced pair of levels 3 P1=2 and 3 P3=2 that arise from the

(a)

D 50,000 (b) 6

2S 2P , 2P 2D , 2D 5 1/2 1/2 3/2 1/2 3/2 40,000 7s 6p 5d 6s 5p 4d 5s 497 8, 498.3 4 4p Energy / cm 30,000 514.9, 515 3 474 8, 475.2 3d 4s 568 3, 568.8 3 615.4, 616.1 –1

Energy / eV 20,000

2 3p

10,000 1 589.0, 589.6 (D)

3s 0 0 2 6 2 6 2 6 2s 2p ns: 2s 2p np: 2s 2p nd: 1 level 2 levels 2 levels Figure 7.10 (a) The line spectrum of sodium. The D-line doublet is not resolved in this image. (b) Schematic Grotrian energy-level diagram for sodium atoms. The transitions giving rise to emission colours are drawn as downward-pointing arrows. The wavelengths emitted are given beside each transition (in nanometres). By far the most intense transitions are those at 589.0 and 589.6 nm that produce a bright yellow doublet – the sodium D lines. The term symbols of all the levels in each column are the same, and the terms 2P and 2D each consist of a pair of closely spaced levels which are not resolved on the energy scale used. (Conversion of the units eV and cm 1 to the SI unit joule (J) is given in Appendix A1.1) 263 Colour from Atoms and Ions

2 2 6 2 configuration 1s 2s 2p 3p before returning to the ground state 3 S1=2 arising from the ground-state configuration of 1s2 2s2 2p6 3s. These two transitions are the only significant lines in the visible spectrum and giveout the familiar yellow sodium light, which makes up approximately 90 % of thevisible emissions. The two 32P levels differ slightly in energy so that the emission consists of two wavelengths, 588.995 and 589.592 nm. These constitute the bright yellow sodium D lines, widely used in spectroscopy and as standard wavelengths at which to record optical properties such as refractive index. Sodium lamps operated at relatively low sodium pressures give a light output dominated by the sodium D lines. This means that objects illuminated by these lamps and observed by reflected light do not show the colour that would be experienced when illuminated by daylight (see Chapter 1). Such lamps are said to have poor colour rendition and have a limited usefulness. To overcome this, high-pressure sodium lamps are now more commonly used in applications such as street lighting, where colour perception is important. The high pressure broadens the lines emitted and other materials, notably mercury (see below), add further emission lines to the spectrum, balancing the D-line emission and giving a light that is perceived as ‘whiter’ (Figure 7.11). Mercury-vapour lights operate in a similar fashion to sodium lights. A high voltage is imposed across a tube containing mercury vapour. Collisions between electrons and mercury atoms excitethem to higher energies and these same atoms emit light as they lose energy again. When the light is first switched on, the low mercury pressure means that only the persistent lines of the spectrum appear and the light has a deep blue colour. As the lamp warms, the pressure increases, more spectral lines appear and these also broaden to give a more white colour, but still with a noticeable blue green aura. The ground-state configuration of mercury, [Xe] 4f14 5d10 6s2, shows that there are a pair of 6s electrons 1 outside of filled inner shells, giving a ground-state level S0. The most important transitions as far as the visible spectrum are concerned are from the excited 6s 7s configuration to the excited 6s 6p configuration (Figure 7.12). These have wavelengths in the blue green (404.7, 453.8 and 546.1 nm), which gives these lamps their rather eerie coloration. The lack of any reds means that faces viewed by reflected mercury light have a pallid appearance. Topartly correct this, mercury tubes are often coated with a fluorescent material which converts the

Figure 7.11 The emission spectrum of a high-pressure sodium street lamp obtained using an inexpensive transmission diffraction grating Colour and the Optical Properties of Materials 264

3 3 1 1 3 S1 P P D2 D 10 80,000 1 6s 7d 3 6s 7d D2 D1,2,3

9 3 1 3 P2 6s 7p D2 D 6s 7p 3 6s 6d 6s 6d 1,2,3 P1 70,000 3 434.7 P0 433 9 579.1 8 577.0 6s 7s Energy / cm 60,000 7 546.1 1 6s 6p P1 Energy / eV 50,000 –1 6 435.8 404.7 3 P2 6s 6p 5 3P 40,000 3 1 P0

4 6sns: 6s np: 6s np: 6s nd: 6s nd: 1 level 3 levels 1 level 1 level 3 levels 30,000 Figure 7.12 Partial Grotrian diagram for mercury atoms. The transitions giving rise to colours are drawn as downward-pointing arrows. The wavelengths emitted are given besides each transition. The most intense line is at 435.8 nm in the blue region of the spectrum. (Conversion of the units eV and cm 1 to the SI unit joule (J) is given in Appendix A1.1)

3 strong ultraviolet emission with a wavelength of 253.65 nm arising from a transition between the 6 P1 level to 1 the ground state 6 S0 (not shown in Figure 7.12) into visible light (see Chapter 9). The spectra of street lights are, in fact, easy to observe or photograph using an inexpensive plastic transmission diffraction grating (see this chapter’s Further Reading).

7.8 Transition Metals and Crystal-Field Colours

The majority of atoms or ions in solids or solutions do not give rise to pronounced colours because the energy difference between the normally occupied ground state and the nearest excited states is generally outside that equivalent to the visible spectrum. The main exceptions to this rule are the enigmatic transition metal ions, which are often described as ‘coloured’. The most important transition metals from the point of view of colour are the 3d transition metals, listed in Appendix A7.1. As an example of these 2 þ 2 þ colours, Figure 7.13 shows aqueous solutions of green Ni(H2O)6 and blue Cu(H2O)6 and crystalline examples of green nickel nitrate Ni(NO3)26H2O and blue copper nitrate Cu(NO3)23H2O. In both solutions and crystals, six water molecules are arranged so that the oxygen atoms form an octahedral coordination polyhedron around a central cation. The colours can be quantiﬁed by recording the absorption spectra of the solutions (Figure 7.14a and b). These show that the nickel-containing solution absorbs in both the violet and red regions of the spectrum, whereas the copper-containing solution absorbs only in the yellow to red region. 265 Colour from Atoms and Ions

2 þ 2 þ Figure 7.13 Crystal-ﬁeld colours of transition metal ions: (a) green Ni (H2O)6 and (b) Cu (H2O)6, both in water solution; (c) green nickel nitrate Ni(NO3)2 6H2O and (d) blue copper nitrate Cu(NO3)2 3H2O crystals. The transition metal ions are in similar environments in both solution and crystals

In these and the other 3d transition metal ions the ﬁve 3d orbitals contain one or more electrons and electron transitions between the various d orbitals are associated with the colours observed. However, a glance at the free-ion terms shows that these do not provide an explanation no suitable energy intervals exist close to the ground state. Thus, the introduction of these cations into solids or liquids must change the energy levels in such a way that transitions that give rise to colours become possible. The way in which this comes about involves the shapes of the d orbitals, which can be described as pointing along or between a set of x-, y- and z-axes Colour and the Optical Properties of Materials 266

(a) 10

2+ Ni(H2O)6

5 Absorption (arbitrary units)

400 500 600 700 Wavelength / nm

(b) 10

2+ Cu(H2O)6

5 Absorption (arbitrary units)

400 500 600 700 Wavelength / nm

2 þ 2 þ Figure 7.14 The absorption spectra of aqueous solutions containing (a) Ni(H2O)6 and (b) Cu(H2O)6 . The absorption scales are arbitrary

(Figure 7.15). The orbitals directed between the axesare the dxy,dyz and dxz set and those pointing along the axes are the dx2 y2 and dz2 pair. In a free ion or atom, all of these orbitals have the same energy. However, this is not true when the atom or ion is placed into a crystal because of the interaction (most easily imagined as repulsion) between the electrons on the surrounding atoms and the d orbitals. If these surrounding electrons were distributed evenly over the surface of a sphere the five d orbitals would still have the same energy as each other, although higher than in the isolated state by an amount E0. If the surrounding electrons are arranged differently, the energy of some of the d orbitals might be different than the others, so that the energies of the orbitals become split (Figure 7.16). This is called crystal-field splitting or ligand-field splitting.3 The extent of the splitting depends upon the symmetry of the surrounding ions and the strength of the local crystal field. The two most important geometries to consider, especially for oxide pigments and ceramics, are octahedral and tetrahedral coordination (Figure 7.17). When an ion is surrounded by an octahedron of

3 The difference between these two labels reflects the method of calculation of the splitting. If the material is treated as ionic and the surrounding charges represented as points (the simplest model), the expression crystal field splitting is appropriate, whereas if molecular orbital theory is used, ligand field splitting is utilized. The terms are usually employed interchangeably and for convenience only the expression crystal field theory will be adopted here. 267 Colour from Atoms and Ions

x x

d dxy xz

y x

d yz dx 2-y 2

dz 2

Figure 7.15 The shapes of the ﬁve d orbitals superimposed upon a set of orthogonal axes. The lobes of electron density in the group dxy,dxz and dyz lie between the axes. The lobes of the pair of orbitals dx2y2 and dx2 lie along the axes

2 negative O ions the d orbitals pointing directly towards the oxygen ions, dx2 y2 and dz2 (the eg pair), will be strongly repelled and so raised in energy compared with those pointing between the oxygen ions, the dxy,dxz 4 and dyz (the t2g group) (Figure 7.18). The labels e and t refer to the degeneracy of the group. A set of orbitals labelled ‘e’ is doubly degenerate; that is, two orbitals with the same energy form the e set. In the same way,

4 The subscript g relates to the symmetry of the atomic or molecular orbitals under discussion. As far as this book is concerned, the subscript g is added when the cation is situated at a centre of an octahedron of surrounding anions, whereas it is omitted when the cation is situated at the centre of a tetrahedron of surrounding anions. Colour and the Optical Properties of Materials 268

(1)

(1) (1) (1)

(5) free ion spherically crystal field d orbitals symmetrical splitting in a field of field rhombic symmetry Figure 7.16 Schematic crystal-fieldsplittingof the energies of the five d orbitals. In a free ion the energies of each of the five orbitals are degenerate (i.e. equal). In a crystal field of spherical symmetry the energies remain degenerate but are increased over that in the free ion by an amount E0. In a field of lower symmetry, the energies of the orbitals split. That shown is the splitting in a field of rhombic symmetry, which totally removes the degeneracy of the set

(a) z

(b) z

Figure 7.17 Acation(largesphere)surroundedby: (a) six oxygen anions (small spheres) arranged as an octahedral anion coordination polyhedron; (b) four oxygen anions arranged as a tetrahedral anion coordination polyhedron. The cation-centred cubic outline indicates that in (a) the anions are located at the cube face centres and in (b) at cube vertices, so that the cation–anion distance is greater in (b) than in (a), leading to a smaller crystal ﬁeld 269 Colour from Atoms and Ions

eg (2 levels)

6Dq(oct) = 3/5 Δ(oct)

t2(3 levels) 4Dq(tet) Δ = 2/5 (tet) 10Dq 10 Dq = E E 0 = Δ(tet) Δ(oct) 0 6Dq(tet) = 3/5 Δ(tet) 4Dq(oct) = 2/5 Δ(oct)

e (2 levels)

t2g(3 levels)

tetrahedral spherical octahedral Δ(tet) = (4/9) Δ(oct)

Figure 7.18 Crystal-field splitting in a field of cubic symmetry. In an octahedral field the t2g set has a lower energy and the eg set a higher energy, while in a tetrahedral field the situation is reversed. The separation of the upper and lower energy levels, the crystal-field splitting, is D(oct) in the case of an octahedral crystal field and (4/9)D(oct) for a tetrahedral crystal field

the label ‘t’ indicates a triply degenerate set, in which case three orbitals with the same energy comprise the group. (Although not relevant to the present situation, it should be mentioned that orbital groups designated ‘a’ consist of a single orbital only.) The crystal-field splitting generates an energy gap between the lower t2g group of orbitals and the upper eg group which is written D or 10 Dq. The distribution of the two sets is unsymmetrical about E0, with the t2g group at 4Dq¼2/5D and the eg group at þ 6Dq¼þ4/5D. When a transition metal ion is surrounded by a tetrahedron of oxygen ions the crystal-field splitting is reversed. In this case the dxy,dxz and dyz orbitals (the t2 group) are raised in energy relativeto dx2 y2 and dz2 (the e pair). The distribution of the two sets is again unsymmetrical about the energy level for a spherically symmetrical distribution of charge E0 with the t2 set at þ 4Dq¼þ2/5D and the e set being at 6Dq¼4/5D (Figure 7.18). The magnitude of the splitting for ions in a tetrahedron will be less than that for ions in an octahedron, and calculations give the result that the tetrahedral crystal field splitting is 4/9 of the octahedral splitting. Note that lower case symbols (t2g, etc.) are used to describe these crystal-field orbitals, as they ignore all electron electron interactions and so behave as ‘one-electron’ states. The colour of a transition metal ion is then supposed to be due to d electrons moving across the relatively small energy gap created by the crystal-field splitting. The magnitude of the crystal-field splitting will depend on the geometry of the surrounding ions and how close they are to the cation. In a strong crystal field, produced when the surrounding anions are close to the cation, the crystal-field splitting is large. This means that the transition energy will be large and any absorption peak will be in the violet or ultraviolet region of the spectrum. In a weak crystal field, produced when the surrounding anions are further away from the cation, the splitting is Colour and the Optical Properties of Materials 270 smaller and any energy peak will be in the red or infrared. This variation accounts for the fact that any particular transition metal cation may exhibit different colours in different compounds, as explained for ruby and emerald below. This neat solution to the question of colour in transition metal ions runs into a problem when the selection rules that apply to transitions are consulted (Section 7.3). It is clearly stated that transitions of the type d to d were forbidden by the Laporte selection rule. However, this rule breaks down for atoms or ions in compounds. The main reason for this is when an ion is not located at a centre of symmetry a degree of mixing betweenvarious orbitals, such as p and d orbitals, can occur. As p to d transitions are allowed, the transitions giving rise to colour are also allowed, to a degree corresponding to the amount of orbital mixing achieved. Thus, ions situated in tetrahedral coordination are not at a centre of symmetry and show quite strong colours. Ions at the centre of a perfect octahedron are at a centre of symmetry, but, in most solids and liquids, thermal agitation of the surroundings and crystal distortions remove the precise symmetry, making transitions possible, though they are often less intense than those from similar ions in tetrahedral sites. Another selection rule is also important. In a free ion, transitions are only allowed between states of the same multiplicity. This rule does not change as an ion is introduced into a crystal, which means that allowed transitions are between states with the same multiplicity. These are called spin-allowed transitions. Transitions between states of differing multiplicity can be weakly allowed, but in general these do not give rise to strong colours. The strength of the crystal field interaction is dependent upon the distance between the surrounding ligands and the central ion. This will vary with temperature, and so the perceived colour of the material will change as the temperature changes, an example of thermochromism. This effect is usually too small to notice over the temperature ranges occurring close to room temperature, but in oxides used as pigments, or in oxide gemstones, the colour at a temperature of several hundred degrees may be quite different to that at lower temperatures. Thermochromism can also come about if temperature causes a change in the geometry of the surrounding ligands. For example, a low-temperature distorted octahedron may transform to a regular octahedral form as the temperature increases and a thermochromic colour change will be registered. Other transformations, from tetrahedral to square planar, for instance, can have a similar thermochromic effect. Having described the source of the colour of transition metal ions, as in the case of free atoms and ions, it is now necessary to construct energy-level diagrams and explain the way in which these vary with the strength of the crystal field in order to explain spectra correctly.

7.9 Crystal Field Splitting, Energy Levels and Terms

7.9.1 Conﬁgurations and strong ﬁeld energy levels

The simplest way to get an idea of the energy levels available to a transition metal ion is to work out the electron configurations assuming that there are no electron electron interactions. This is called the strong field approach, and basically assumes that the crystal field is so strong that it dominates all other interactions, such as electron electron repulsion. Electrons are then simply allocated to the split orbitals, filling the lower energy orbitals first. Taking octahedral geometry as an example, suppose that the ion has n d electrons in total, of which (n p) are in the lower t2g set and p are in the upper eg set. The energy of the ion E (with respect to the spherically symmetrical charge distribution E0) is computed by noting that the energy of the t2g orbitals set is 4 Dq and that of the eg set is þ 6 Dq. The energy of an ion is then:

E ¼½ðnpÞð4ÞþðpÞð þ 6Þ Dq 271 Colour from Atoms and Ions

To illustrate this, the energy levels of a 3d3 ion in an octahedral site are obtained in the following way. The lowest energy for the ion will be when all three d electrons are in the lowest energy levels, so that this 3 0 configuration can be written (t2g) (eg) , and the energy is 12 Dq. The next lowest energy for the ion will 2 1 correspond to two electrons in the t2g orbitals and one in the eg pair, (t2g) (eg) , giving an energy2 Dq. Another 1 2 energy level will correspond to the electron distribution, (t2g) (eg) , lying at þ 8 Dq. Finally, the highest energy 0 3 3 will correspond to all electrons in the upper level, (t2g) (eg) ,at þ 18 Dq. The 3d configuration will then give rise to four strong-field energy levels. Energy levels for other d-electron populations or for ions in a tetrahedral crystal field can be calculated in a similar fashion. On the basis of this model, the spectrum of a transition metal ion in an octahedral crystal field will consist of a 2 þ set of peaks, with energies given by multiples of 10 Dq. The spectrum of Ni(H2O)6 (Figure 7.14a) does not agree with this simple prediction. The three peaks in the spectrum are at wavelengths of 395 nm, 680 nm (centre) and 1176 nm. These are at energies of 3.14 eV (5.03 10 19 J), 1.82 eV (2.92 10 19 J) and 1.05 eV (1.69 10 19 J). Taking 10 Dq as 1.05 eV, although the highest of the energy values (3.14 eV) is close to 3 1.05 eV,the middle value does not fit at all. This indicates that the simple model needs refinement. If some degree of electron electron interaction is introduced (i.e. the crystal field becomes weaker), then these levels will split, just as a configuration such as 3d2 splits into a number of terms of varying energy when electron electron repulsion is considered (Figure 7.5).

7.9.2 Weak ﬁelds and term splitting

At the other extreme, it is possible to assume that the crystal field is very weak. Electron electron repulsion is then most important. In fact, in the case of a zero-strength crystal field, the dominant set of energy levels are described by term symbols (Section 7.2). As the strength of the crystal field increases, this will modify the energies represented by the terms in a predictable fashion, first set out by Bethe in 1929. The result is that for ions in a field of cubic symmetry, which is applicable to octahedral and tetrahedral crystal fields, S and P terms remain as single energy states, a D term splits into two energy states, an F term into three and a G term into four (Table 7.7). The multiplicity of the term is carried over onto the new states. The energy states in the crystal are given labels that parallel those for single-electron orbitals mentioned in the previous section and describe the degeneracy of the orbitals. Thus, an S term, which is singly degenerate, is labelled A in a crystal field, a P term, which is triply degenerate, is labelled T, meaning triply degenerate, a D term splits into two, a doubly degenerate E term and a triply degenerate T term, and so on. Different configurations of sets of orbitals with the same degeneracy are labelled with a subscript 1, 2 and so on. Thus, a P term gives rise to a T1 term, while a D term gives rise to a T2 term. The subscript ‘g’ means a

Table 7.7 Splitting of terms in ﬁelds of cubic symmetry

Terms in tetrahedral Terms in octahedral Free ion term crystal ﬁeld crystal ﬁeld

SA1 A1g PT1 T1g DE,T2 Eg,T2g FA2,T1,T2 A2g,T1g,T2g GA1,E,T1,T2 A1g,Eg,T1g,T2g Colour and the Optical Properties of Materials 272 centre of orbital symmetry exists, as described in the previous section. (Further information is given in the Further Reading.) The splitting of the ground-state terms of the 3d transition metal ions in an octahedral crystal field can now be drawn (Figure 7.19a). It is seen that the arrangement of the new energy states is symmetrical about the d5 6S term. This comes about because orbitals with n electrons can be treated as mirrors of those containing n holes. Thus, the term splitting of a d1 ion (an empty shell plus one electron) is equivalent to that of a d9 ion because the latter configuration can be regarded as equivalent to a filled shell plus one hole. However, the order of the states is reversed. The energies of the states can also be calculated for a weak crystal field. These give results similar to those in a strong field, but with significant differences for some terms (Table 7.8). It is also found that if the order of the split terms is inverted then the arrangement appropriate to ions in a 2 tetrahedral field is obtained (Figure 7.19b). Thus, if an F state for a d ion splits in an octahedral field into A2g, T2g and T1g terms (in decending order), then in a tetrahedral field it will split into T1,T2 and A2 terms (in decending order). The energy of the splitting will remain the same, in units of Dq, but the magnitude of the

4T (a) 2 1g Eg 3 A2g 5 T2g

2 3 4 5 6 6 D F F 4 D S A1g 3 T2g T2g

2 T2g 5 Eg 4 A2g 3 T1g d 1 d 2 d 3 d 4 d 5

3 T1g 5E g 4 A2g 2 T2g

5 4 3 2 D F F 3 D 4 T2g T2g

5 T2g 2 Eg 3A 4 2g T1g d 6 d 7 d 8 d 9 Figure 7.19 Splitting of the ground-state terms of d1 to d9 ions in a field of cubic symmetry (schematic): (a) octahedral field; (b) tetrahedral field. The magnitude of the splitting in an octahedral field will be approximately double that in a tetrahedral field 273 Colour from Atoms and Ions

4A (b) 2 1 T2 3 T1 5E

2 3 4 5 6 6 D F F 4 D S A1 3 T2 T2

5 T2 4T 3 1 A2 d 1 d 2 d 3 d 4 d 5

3A 5 2 T2 4 T1 2E

5D 4F 3F 2D 4 3T T2 2

5E 2T 3 2 T1 4 A2 d 6 d 7 d 8 d 9 Figure 7.19 (Continued) splitting will be smaller for tetrahedral ﬁelds compared with octahedral ﬁelds because of the difference between Dq(octahedral) and Dq(tetrahedral).

7.9.3 Intermediate ﬁelds

The separation of each of the split levels described in the previous section will increase as the magnitude of the crystal field increases. Ultimately these must link up with the energy levels described in the strong-field case. In most real crystals the energy levels are then somewhere between the weak- and strong-field extremes. The connection can be described schematically as:

n 2S þ 1 2S þ 1 n p p n d ! L ! A; E; T t2g eg d

2 þ 2 þ With this information it is now possible to interpret the colour of green Ni(H2O)6 and blue Cu(H2O)6 . Take the copper case first. The zero-field ground state of the d9 Cu2 þ ion is 2D. When the ion is introduced into a crystal field, the energy of the 2D term will be greater than the equivalent energy of an ion in a vacuum by the 2 interaction energy E0 (Figure 7.16) but will still remain as a precise D term if the crystal-field energy is set at Colour and the Optical Properties of Materials 274

Table 7.8 Energies of terms in an octahedral crystal ﬁeld

Free ion electron Free ion Term in an conﬁguration ground state term octahedral ﬁeld Energy/Dq 1 2 2 d D T2g 4 2 þ Eg 6 2 3 3 d F T1g 6 3 T2g þ 2 3 A2g þ 12 3 4 4 d F A2g 12 4 T2g 2 4 T1g þ 6 4 5 5 d D Eg 6 5 T2g þ 4 5 6 6 d S A1g 0 6 5 5 d D T2g 4 5 þ Eg 6 7 4 4 d F T1g 6 4 T2g þ 2 4 A2g þ 12 8 3 3 d F A2g 12 3 T2g 2 3 T1g þ 6 9 2 2 d D Eg 6 2 T2g þ 4

2 2 2 zero. As the crystal field increases, the D term will divide into Eg and T2g levels in the octahedral field of the surrounding water molecules; more precisely, that of the surrounding oxygen ions. The divergence is a linear 2 þ 2 function of Dq, with slopes of 6Dq( Eg) and 4Dq( T2g) (Figure 7.20). This representation is called an 2 Orgel diagram. At very strong fields the ground-state Eg level is associated with the ‘one-electron’ 6 3 configuration t2g eg obtained by placing six of the nine d electrons into the lowest energy t2g orbitals first and then allocating the remaining three to the eg levels. In the excited state, one electron is promoted from the 5 4 2 lower to the upper set to give the configuration t2g eg associated with the T2g state. This indicates that the spectrum should contain one absorption peak. The maximum is found to occur at 780 nm, in the near infrared, 5 2 2 corresponding to the transition between the upper and lower energy level, T2g Eg (Figure 7.21). The energy separation should be equal to 10 Dq, with a value of 1.6 eV (2.55 10 19 J). However, a word of caution is needed. The peak itself is very broad and suggests that it is necessary to take into account interactions not included so far. (In fact, it is found that the coordination polyhedron is not a regular octahedron, but is distorted so that the symmetry changes to tetragonal, which accounts for this feature.) 2 þ 8 The colour of Ni(H2O)6 can be explained in an analogous fashion. The zero-field ground state of the d 2 þ 3 3 3 3 3 Ni ion is F. As the crystal field increases, the F term divides into A2g, T2g and T1g in the octahedral field of the surrounding water molecules. These should show a linear dependence upon Dq, with slopes of

5 A spectroscopic transition is conventionally written with the higher energy state (H) ﬁrst and the lower energy state (L) second. This means that an absorption of energy is written H L, and an emission of energy is written H ! L. This is adopted when the equations describing the transition are not written in a ‘chemical equation’ format. 275 Colour from Atoms and Ions

2 T2g Energy 2D 10 Dq

2 Eg

Dq þ Figure 7.20 Orgel diagram (schematic) for a d9 ion such as Cu2 in an octahedral ﬁeld. The separation of the two energy levels is equal to 10 Dq

Absorption

400 nm: violet

blue-green transmission

700 nm: red 780 nm: infrared absorption 3 T2g Energy / eV

0 2 Eg þ Figure 7.21 The electronic transition responsible for the absorption spectrum of Cu2 ions in water solution and 2 þ blue hydrate crystals, Cu (H2O)6 Colour and the Optical Properties of Materials 276

3 3 3 12 ( A2g), 2( T2g) and þ 6( T1g). However, the energy-level diagram for the free ion shows that the terms 1D, 3P and 1G are quite close in energy to the 3F term, and these must also be considered. As optical transitions are only expected between the triplet terms, 1D and 1G can be omitted in the first instance. (Note that this is not always so. The operation of the ruby laser depends upon a spin-disallowed transition; Section 7.11.) However, 3 3 the P term cannot be ignored and transforms into a T1g term in the octahedral crystal field with the slope of the 3 3 linear dependence equal to zero. This means that, at moderate field strengths, the upper T1g level from the F term will cross it (Figure 7.22a). Now the non-crossing rule of quantum mechanics states that two energy states with the same symmetry arising from a single ion never cross. Thus, the Orgel diagram for a d8 ion shows these two straight lines becoming curved, as if repelling each other (Figure 7.22b).

(a) 3 T1g 3 3 P T1g

8 Dq 3F Energy 3 T2g

10 Dq

3A Dq 2g

3 T1g (b)

3P 3 T1g

8 Dq - interaction energy 3F Energy 3 T2g

10 Dq

3A Dq 2g þ Figure 7.22 Orgel diagram (schematic) for a d8 ion such as Ni2 in an octahedral ﬁeld: (a) without term 3 3 interactions between the two T1g levels (crossing allowed); (b) with interaction between the two T1g levels 3 3 (non-crossing rule applied). The separation of the two lower energy levels ( A1g and T2g ) is equal to 10 Dq in both cases 277 Colour from Atoms and Ions

The spectrum should contain three peaks that correspond to the transitions and energies

3 3 19 1 1. T2g A2g 1176 nm (1.05 eV, 1.69 10 J, 8503 cm ) ¼ 10 Dq 3 3 3 19 1 2. T1g (from F) A2g 680 nm (1.82 eV, 2.92 10 J, 14 706 cm ) ¼ 18 Dq interaction energy 3 3 3 19 1 3. T1g (from P) A2g 395 nm (3.14 eV, 5.03 10 J, 25 316 cm ).

Only two of these are close to the visible, one at each end of the spectrum, in good agreement with the absorption data. The energy separation for the lowest energy transition, in the infrared, is seen to be equal to 10 Dq (¼D) for this chemical environment. The energy separation for the next lowest transition ideally corresponds to 18 Dq. However, the non-crossing rule decreases this by an amount of energy called the interaction energy or conﬁguration interaction energy. The third absorption peak is not directly related, in this simple model, to the 2 þ crystal-ﬁeld splitting, Dq, and so provides no new information on this parameter. The colour of Ni(H2O)6 is due to the two transitions:

3 3 3 T1g (from F) A2g 680 nm 3 3 3 T1g (from P) A2g 395 nm

The absorption of energy in the near ultraviolet and in the far red remove blue and red from the transmission spectrum, resulting in the green colour perceived (Figure 7.23). However, it is obvious that the central peak of the spectrum is split into two components, again indicating that a further reﬁnement of the simple theory so far presented is needed. (In fact, it is necessary to take the coupling between the spins and the orbital angular momentum, or spin orbit coupling, into account for this.)

7.10 The Colour of Ruby

Ruby consists of single crystals of aluminium oxide (a-Al2O3) containing about 0.5 % Cr as an impurity. The Cr3 þ impurity ions in ruby are distributed at random over some of the positions normally reserved for Al3 þ in the oxide structure. The formula of the gemstone can be written (CrxAl1 x)2O3. When x ¼ 0 (pure Al2O3) the stone is colourless and found as the mineral corundum. At very small values of x close to 0.005 the crystal is coloured a rich ruby red. As the Cr3 þ concentration increases, so the colour becomes grey and then dull green, which is the colour of pure Cr2O3, used as the pigment chrome green. 3 þ The fact that ruby is coloured while Al2O3 is colourless indicates that it is the Cr in the structure that is of paramount importance. The colour changes can be explained in terms of crystal-field splitting of the energy 3 þ 3 þ levels of the Cr ion. The oxides Al2O3 and Cr2O3 are isostructural and the positions occupied by Cr all across the composition range are at the centres of slightly distorted octahedral sites formed by six nearest neighbour oxygen ions. The outer electron configuration of the Cr3 þ ion is 3d3 and the ground-state term is 4F. This splits in an 4 4 4 4 octahedral crystal field to give the states A2g at 12 Dq, T2g at 2 Dq and T1g at þ 6 Dq. A higher P state 4 8 2 þ gives rise to a further T1g level. Apart from the multiplicity, the diagram is similar to that for the d ion Ni (Figure 7.22) and three absorption bands are expected. These are

4 4 19 1 1. T2g A2g 556 nm (2.23 eV, 3.58 10 J, 18 000 cm ) ¼ 10 Dq 4 4 4 19 1 2. T1g (from F) A2g 400 nm (3.10 eV, 4.97 10 J, 25 000 cm ) ¼ 18 Dq interaction energy 4 4 4 19 1 3. T1g (from P) A2g 270 nm (4.59 eV, 7.35 10 J, 37 000 cm ). Colour and the Optical Properties of Materials 278

Absorption

395 nm: violet absorption 3T 3 1g

green transmission

2 3T 1g 680 nm: red absorption Energy / eV

3T 1 2g

0 3 A2g þ Figure 7.23 The electronic transitions responsible for the absorption spectrum of Ni2 ions in water solution, 2 þ Ni (H2O)6

The ﬁrst two of these transitions contribute to the colour of the gemstone. The absorption at 556 nm removes green yellow and the absorption at 400 nm removes violet. Between the absorption curves there is a relatively small blue transmission window at 680 nm and a red transmission window is present at wavelengths greater than about 650 nm. This means that the colour transmitted by the ruby will be red with something of a blue purple undertone. The colour of ruby, however, is richer than this explanation suggests, and more detail needs to be added. Two additional factors need to be taken into account: spin-forbidden transitions and the crystal symmetry. The transitions that give rise to colour, such as those detailed above, are forbidden in terms of parity but are spin allowed because the multiplicity of all the terms is identical and equal to four. The free ion terms of Cr3þ show that a 2G term is found only slightly higher in energy than the 4P term (Figure 7.24). A G term splits into four in an octahedral crystal ﬁeld (Table 7.7) and in ruby two of these new levels fall within the spread of the levels derived from the 4F ground state (Figure 7.24). 2 The most important of these, from the point of view of colour, is the lowest energy Eg level. Direct excitation 4 2 from the ground-state A2g to the Eg level is forbidden by both the parity and multiplicity selection rules. However, a different circumstance operates for the excited states. At the same time as excitation is occurring, 279 Colour from Atoms and Ions

4 T1g 2 A1g

4 T1g

2 T2g

4 T2g

2 T1g 2 Eg 4F

2 A2g

Free ion Octahedral crystal term field þ Figure 7.24 Energy levels of the 3d3 ion Cr3 due to splitting of free ion terms 4F,4P and 2G in an octahedral crystal field (schematic) manyof the higher energy Cr3 þ ions return to theground state by emitting exactly the same amount of energy as 4 4 was absorbed, so as to drop back to the ground state from either T1g or T2g. However, as these transitions are forbidden by the parity selection rule, they are not fast. Some ions lose energy instead to the crystal structure, warming it slightly, dropping back only to the 2E energy level. This is not an optical transition, but involves heat energy, phonon exchange, and so is not bound by the selection rules given earlier for optical transitions. It is described as a radiationless or phonon-assisted transition. (However, the selection rules make it clear that a direct (optical) transition from the ground state to the 2E state by absorbing energy is low, and so 2E only becomes filled by this roundabout process.) The ions in the 2E state also slowly lose energy and return to the ground state. This transition gives rise to red light emission, which features as a narrow band, called the intercombination band J1 or R, close to 693 nm, in the ruby spectrum. This fluorescent radiation (see Chapter 9) enhances the colour of the best rubies. At compositions close to Cr0.005Al0.995O3 it can be made to dominate light emission, and the result is laser action (Section 7.11). The second feature that adds to the colour of ruby stems from the symmetry of ruby crystals. The crystal structure of ruby is trigonal (but usually referred to hexagonal axes), and as with all crystals of symmetry lower than cubic, the absorption spectrum depends upon the polarisation of the light used for the illumination. Ruby crystals are dichroic (Section 4.8). In ruby, two absorption spectra arise: one for light polarised parallel to the crystallographic c-axis and one for light polarised perpendicular to the c-axis (Figure 7.25). Although these spectra are very similar to each other, noticeable differences in colour are apparent when ruby crystals in Colour and the Optical Properties of Materials 280

4 Absorption

violet 4 4A T1g 1 absorption 3 4E ~ 400 nm

4 4T A2 2g yellow/green/orange 4E absorption ~ 556 nm 2 2 Eg deep red fluorescence

Energy / eV polarisation parallel to c-axis (e-ray): orange-red colour 1

polarisation perpendicular to c-axis (o-ray): purple-red colour

0 3 A2g Figure 7.25 The electronic transitions that give rise to dichroism in ruby. The new energy levels are produced by the decrease in symmetry from octahedral (Figure 7.24) to rhombic in crystals, which causes a splitting of the 4T levels. Light polarised perpendicular to the optic axis (the o-ray) gives the gemstone a purple–red colour, while 2 light polarised parallel to the optic axis (the e-ray) gives an orange–red colour. The Eg level also splits, but the scale of the diagram is too small to show this differing orientations are observed in polarised light. In ruby, when the plane of polarisation of the light is perpendicular to the c-axis the crystal is perceived as ruby coloured, but when it lies parallel to the c-axis the crystal takes on a more orange hue. 4 4 4 In terms of crystal-ﬁeld theory, the excited T2g and T1g (from F) are split due to the change of local symmetry from cubic (in an ideal octahedron) to trigonal (point group C3v) in the crystal (Figure 7.25) as follows:

4 4 4 T2g ! A2 and E 4 4 4 4 T1g (from F) ! A1 and E

The selection rules now mean that light polarised parallel to the c-axis (the e-ray) interacts with only the 4A levels and light polarised perpendicular to the c-axis (the o-ray) with the 4E levels. The separation of the new 281 Colour from Atoms and Ions levels is small, of the order of 0.06 eV (500 cm 1), but does give a colour change that is readily detected by eye. When illuminated by light polarised perpendicular to the c-axis (the o-ray) the ruby is purple red (ruby) coloured, while when illuminated by light polarised parallel to the c-axis (the e-ray), the colour is perceived as orange red. 2 In addition to these effects, the E level giving rise to the intercombination band J1 (R) is also found to be split into two components due to the same symmetry change, so that the red line at 693 nm is resolved into two narrow lines, R1 at 693.5 nm and R2 at 692.3 nm (not differentiated at the scale of Figure 7.25).

7.11 Transition-Metal-Ion Lasers

7.11.1 The ruby laser: a three-level laser

The first laser constructed was the ruby laser, built by Maiman in 1960. It consisted of a ruby crystal (Cr0.005Al1.995O3) about 7 cm long. One end face was silvered to give total reflection while the other end was partially silvered so as to release any stimulated emission. The population inversion was created by a bright flash of light from a xenon flash tube. The whole was surrounded by a reflecting shield (Figure 7.26). Laser operation makes use of the electronic energy levels of Cr3 þ in a crystal field described above. The electron transitions which lead to colour in rubies are due to the transitions

4 4 1. T2g A2g 556 nm, absorbs yellow green 4 4 2. T1g A2g 400 nm, absorbs violet.

These transitions are forbidden in terms of parity but are spin allowed. In addition, account must be taken of 2 4 2 the E term. Direct excitation from the ground-state A2g term to the E term is forbidden by both the parity and multiplicity selection rules. As described in the previous section, this level is ﬁlled by a roundabout process that involves the shedding of energy to the crystal structure as a radiationless transition. The rates of the transitions between these states are

4 4 5 1 1. T2g ! A2g:3 10 s 4 2 7 1 2. T2g ! E: 2 10 s .

The second of these two transitions is about 100 times faster than the ﬁrst. The rates of the transitions from the 4 2 4 T1g energy level to E and A2g are of a similar magnitude. This means that, on irradiating the ruby with white light, a signiﬁcant number of atoms end up in the 2E state. For the same quantum mechanical reasons that forbid

reflector

Xenon flash tube

ruby crystal laser output

Figure 7.26 The original ruby laser (Maiman, 1960) (schematic) Colour and the Optical Properties of Materials 282 the direct transition from the ground state to the 2E state, the transition from the 2E level back to the ground state is also forbidden, and so atoms in the 2E state have a long lifetime. Thus, it is possible to build a population 2 4 inversion between the E and A2g levels. Laser operation takes place in the following way. An intense flash of white light is directed onto the crystal. 3 þ 4 4 This process is called optical pumping. This excites the Cr ions into the T2g and T1g states. These then lose energy by radiationless transitions and ‘flow over’ into the energy level 2E. If the initial flash is intense enough 2 4 it will cause a population inversion between E and A2g. About 0.5 ms after the start of the pumping flash, some spontaneous emission will occur from 2E. In order to prevent these first photons from escaping from the crystal without causing stimulated emission from the other excited ions, one end is coated with a mirror and the other with a partly reflecting mirror. In this case the photons are reflected to and fro, causing stimulated emission from the other populated 2E levels. Once started, the stimulated emission rapidly depopulates these levels in an avalanche. There will be a burst of red laser light of wavelength 694.3 nm which emerges from the partly reflecting surface. In the original laser, silver mirrors were used. However, silver absorbs as much light as it transmits. This resulted in overheating, which caused crystals to deform or crack. Now the mirrors are thin-film dielectric mirrors, one of which transmits about 1 % of the incident photons. Following the light burst, the upper levels will be empty and the process can be repeated. The ruby laser generally operates by emitting energy in short bursts,each of which lasts about 1 ms. This is referred toas pulsed operation. The ruby laser is called a three-level laser, because basically three energy levels are involvedin the operation. 4 4 4 These are the ground state ( A2g), an excited state reached by optical absorption or pumping ( T2g or T1g) and an intermediate state of long lifetime (2E) reached by radiationless transfer from the optically accessible state and from which stimulated emission (laser emission) occurs to the ground state. It is energetically costly to obtain a population inversion in a three-level laser because one must pump more than half the population of the ground state to the middle level. Moreover, very little of the electrical energy supplied to the flash lamp ends up pumping photons, and carefully designed reflectors are essential. Finally, the energy lost in the transitions from 4 4 2 T2g and T1g to E ends up as lattice vibrations, which cause the crystal to heat up considerably. To make sure that the ruby does not overheat and shatter, it is necessary to cool the crystal and to space the pulses to allow the heat to dissipate.

7.11.2 The titanium–sapphire laser

The titanium sapphire laser is, at present, the laser chosen for the generation of femtosecond pulses. Chemically, it is very similar to the ruby laser, in that the laser medium is a crystal of corundum doped þ with a small amount of Ti3 . This ion has a 3d1 configuration and the impurities are located in octahedral sites in the corundum structure, totally analogous to Cr3þ in ruby. In an octahedral crystal field the 2D free-ion state 2 2 3þ splits into a T2g ground state and a Eg excited state (Figure 7.27). The absorption spectrum of Ti ions in 2 2 corundum should then consist of a single maximum corresponding to the transition Eg T2g and the 2 ! 2 emission spectrum, due to the transition Eg T2g, should be similar. This transition is generally regarded as the operative laser transition. In reality, the situation is slightly more complex. The symmetry of the Ti3þ sites in corundum is not perfectly 2 2 octahedral, causing both the ground-state T2g energy level and the upper Eg level to split (Figure 7.27). This results in the absorption spectrum showing two overlapping peaks, at wavelengths of approximately 475 and 550 nm. The laser transition, from the lower of these levels to the ground-state levels is at a wavelength of 2 ! 2 approximately 800 nm. (Note, however, that the transition is still labelled as Eg T2g in most laser literature.) These are normal crystal-field-generated energy levels, and the excited and ground states have the same multiplicity. This means that the lifetime of the excited state is short (approximately 3.2 ms) and spontaneous 283 Colour from Atoms and Ions

2 ~475 nm E2g

10 Dq 2D ~550 nm ~800 nm

2 T2g

free ion octahedral corundum field

3þ 1 Figure 7.27 Schematic energy-level diagram for Ti (3d ) ions in corundum (Al2O3). To a ﬁrst approximation the free-ion term splits into two, separated by 10 Dq. In corundum, distortion of the coordination polyhedra surrounding the cations splits both the ground and excited states, approximately as shown

emission is the expected mechanism of energy loss. To obtain a population inversion it is necessary to pump the crystal with an intense laser beam. Dye lasers (Section 8.12), frequency-doubled Nd:YAG lasers (Section 7.16) and other lasers have been used for this excitation.

7.12 Emerald, Alexandrite and Crystal-Field Strength

The effect of the strength of the crystal field on colour is demonstrated by comparing ‘ruby-red’ ruby with the gemstone emerald, with a characteristic ‘emerald green’ hue. Emeralds possess the hexagonal beryl (Be3Al2Si6O18) structure, which, when pure, is a colourless mineral. The structural framework is composed of Si6O18 rings forming tunnels parallel to the c-axis linked by Be-centred oxygen tetrahedra and Al-centred þ þ octahedra. As with ruby, a trace of Cr3 substitutes for some Al3 . The source of colour in both gemstones is thus Cr3þ in octahedral sites. The energy-level diagram for ruby is therefore relevant (Figure 7.24). However, in beryl the octahedra surrounding the Cr3 þ ions are slightly larger than in corundum and so the crystal field experienced by the Cr3þ in emerald is weaker than in ruby. The energy-level diagram remains essentially the 4 4 4 same, but there is a shift in the energy levels T1g and T2g towards the ground-state level A2g. This causes the two main absorption bands to move towards the lower energy red end of the spectrum. The band that absorbs yellow/green in ruby (556 nm) now absorbs yellow/red with a peak at 650 nm. The violet-absorbing band in ruby (400 nm) now absorbs more blue, with a peak at 450 nm. Between the absorption curves there is a blue green transmission window at 500 nm. Emeralds, therefore, absorb red and some blue and transmit green with some residue of blue to give the typical emerald colour. The crystal structure of emerald is hexagonal, and just as with ruby, crystals are dichroic; the colour depending upon the direction of the polarised light which irradiates them. In addition, the 2E state gives a red fluorescence just as with ruby. However, neither of these effects is as noticeable as in the case of ruby itself. Alexandrite is an extremely rare mineral. In daylight or the light from a fluorescent tube the stone looks blue green, while in incandescent light from an ordinary tungsten-filament lamp or candlelight the stone appears deep red. The gemstone is a form of the mineral chrysoberyl, BeAl2O4. The crystal structure is orthorhombic, Colour and the Optical Properties of Materials 284 of the olivine structure type. The oxygen atoms in the structure are in approximately hexagonal close packing, with the Al atoms occupying octahedral positions and the Be atoms in tetrahedral positions. As with ruby and emerald, the colour of alexandrite is due to a small amount of Cr3þ impurity ions replacing Al3þ ions in the octahedral sites of the structure. The colour, therefore, is produced in the same way as in emerald and ruby, by the crystal-field splitting of the energy levels arising from the 3d3 configuration of the Cr3þ impurity ions and the energy-level diagram in Figure 7.24 remains valid. In alexandrite, the magnitude of the crystal field is about halfway between that of ruby and emerald. The two important absorption peaks lie midway between those found in these latter gemstones. However, to obtain the true alexandrite colour effect the concentration of Cr3 þ must be such that the two absorption bands, one in the blue region and one in the yellow/ red region of the spectrum, are equal; and just as importantly, thewindows of transmission of red and green light are also comparable. In these especial circumstances the colour of thegemstone that is perceivedby the eye then depends upon the spectral characteristic of the light falling on the stone. In daylight or slightly blue-rich light, to the eye, which is more sensitive to green than to red, the stone takes on the appearance of emerald. On the other hand, if this incident light is rich in red, as in the case of light from an incandescent source, little green light is returned to the eyeandthe stone looks a ruby colour. That is to say, the colour noted is due tothe perception of the relative amounts of red and blue green light reaching the eye, not due to any changes in the crystal field under different types of illumination. For the alexandrite effect to be seen, the impurity content and the crystal field must be finely matched. It is for this reason that natural alexandrite is rare. However, corundum (aluminium oxide, Al2O3) can be doped to produce synthetic pseudo-alexandrites, which are sometimes sold as real alexandrite by unscrupulous dealers. Note that because alexandrite (and chrysoberyl) possesses an orthorhombic crystal structure it has two optic axes and is a biaxial material. The crystals exhibit strong trichroism when observed in transmitted linearly polarised light. The colours vary from purple red to orange and green, depending upon the relative orientation ofthe crystal and the plane of polarisation of the light. This is not related to the alexandrite effect and is shown by colourless chrysoberyl as well as by alexandrite itself.

7.13 Crystal-Field Colours in Minerals and Gemstones

The colour produced by a transition metal ion will depend on the local crystal field, as the previous sections, pertaining to Cr3þ in octahedral sites, make clear. A further example of this variation is given when this latter ion is incorporated into chrome alum. The colour perceived is purple, again due to crystal-field changes, as the Cr3þ is now surrounded by six water molecules. When absorption bands due to crystal-field splitting occur well within the visible spectrum, then even small crystallographic changes lead to sufficiently large changes in the local crystal field at the transition metal ion that the perceived colours change enormously. Despite this, many transition metal ions are thought of as showing a typical colour. Copper compounds, for example, are usually green-blue and Fe2 þ imparts a pale watery green colour to oxides and hydrates. Taking copper as an example, in copper oxides, hydroxides or hydrates the Cu2 þ ions are usually coordinated by six oxygen atoms in a distorted octahedral coordination polyhedron. Although the crystal field varies from one compound to another, the absorption peak usually lies in the infrared. The perceived colour, blue or blue green, is due to the intrusion of this absorption band into the red end of the visible spectrum. The exact position of the peak is thus largely irrelevant, with colour changes being due to the small differences in encroachment into the visible. Thus, these compounds will always appear to be blue green. The exact tone will depend upon both the crystal-field strength and the consequent encroachment, which gives ample room for subtle colour variation. Figure 7.28 shows a sample of the mineral malachite, Cu2(OH)2(CO3), with a green colour. This is mixed with the related mineral azurite, Cu3(OH)2(CO3)2, which is bright blue. Although the chemical formulae 285 Colour from Atoms and Ions

2 þ Figure 7.28 The differing crystal field colours of Cu ions in malachite (Cu2(OH)2(CO3), green) and azurite (Cu(OH)2(CO3)2, blue) intermingled in a mineral sample of these two minerals are similar, the crystal field in azurite is sufficiently different to that in malachite that quite different colours are perceived: blue versus green. Many ions can occur in more than one type of coordination and important variations of colour can occur. An ioninatetrahedralsitewillgenerallyexperienceacrystalfieldabout4/9ofthatexperiencedbythesameioninan octahedral field. Thus, absorption peaks will move from the violet towards red. In the case of octahedrally coordinated Cu2 þ , which has typically blue colours due to absorption in the red, an ion in a tetrahedral site will now have an absorption peak well into the infrared, and hence tetrahedral Cu2 þ will not show the typical blue colour. A similar change in coordination leads to the colour change exhibited by many moisture indicators or drying agents. These are materials which often contain Co2 þ ions. In the dry state these ions are in tetrahedral coordination and appear deep blue the same colour as cobalt in glass. When these materials pick up water, the 2 þ coordination changes to octahedral and Co (H2O)6 units form. In these, the oxygen atoms in the water molecules surround the Co2 þ ions in an octahedral configuration. The colour is now pink. Despite the variations in crystal-field colours, it is useful to list some of the more characteristic colours of the 3d transition metal compounds. These are given in Table 7.9. As was highlighted by ruby and emerald, many gemstones are actually coloured by the presence of small amounts of transition metal impurities in what would otherwise be colourless crystals. A short list of some gemstones coloured by transition metal ion impurities is given in Table 7.10. Colour and the Optical Properties of Materials 286

Table 7.9 Typical colours shown by 3d transition metal compounds

Number of d Approximate Ion electrons Colour symmetry Examples þ a Ti3 1 purple tetrahedral doped glass þ V4 1 red tetrahedral doped glass þ V3 2 green tetrahedral doped glass blue octahedral Al2O3 þ Cr3 3 green tetrahedral doped glass green octahedral emerald (Be3Al2Si6O18) red octahedral Al2O3 (ruby), doped TiO2 violet purple octahedral chrome alum (KCr(SO4)3 12H2O) þ Mn3 4 purple tetrahedral doped glass þ Mn2 5 yellow tetrahedral doped glass red octahedral MnCO3 green octahedral MnO pink octahedral MnSiO3 þ Fe3 5 yellow green tetrahedral doped glass red brown octahedral Fe2O3, rust 2þ 2þ Fe 6 blue green octahedral Fe(H2O)6 in solution and hydrates 2 þ Co 7 blue tetrahedral CoAl2O4, glass 2þ pink octahedral Co(H2O)6 in solution and hydrates 2þ 2þ Ni 8 green octahedral Ni(H2O)6 in solution and hydrates, NiO yellow octahedral doped Al2O3, NiCl2 2 þ Cu 9 green octahedral Cu2(OH)2(CO3); malachite 2þ blue octahedral Cu(H2O)6 in solution and hydrates a Glass refers to silicate glass.

Table 7.10 The colours of some gemstones

Structure Origin of colour and Gemstone General formula type Colour cation replaced 2þ Garnet Ca3Al2Si3O12 garnet red Fe in cubic (8 coordinate) þ Ca2 site 2þ 2þ Peridot Mg2SiO4 olivine yellow Fe in octahedral Mg site green 3þ 3 þ Topaz Al2SiO4(OH) topaz yellow Fe in octahedral Al site 3þ 3 þ Emerald Be3Al2Si6O18 beryl green Cr in octahedral Al site 3þ 3 þ Alexandrite BeAl2O4 olivine red/green Cr in octahedral Al site 3þ 3 þ Ruby Al2O3 corundum red Cr in octahedral Al site 2þ 3þ Rubellite CaLi2Al7(OH)4(BO3)3Si6O18 tourmaline pink red Mn in octahedral Al site 2þ 3 þ Indicolite CaLi2Al7(OH)4(BO3)3Si6O18 tourmaline blue Fe in octahedral Al site 2 þ 2 þ Turquoise CuAl6(PO4)4(OH)8 4H2O turquoise blue Cu in octahedral (Cu ) site green 287 Colour from Atoms and Ions

7.14 Colour as a Structural Probe

The spectrum of a transition metal ion in a solid can give information about the local position of the ion because the colour depends upon the crystal ﬁeld. Consider as an example the problem of cation distribution in oxide spinels. The spinel structure is adopted by many compounds with a formula AB2X4, where A and B are medium- sized cations and X represents an anion, most often O2 . In this structure the anions are in a close-packed array and the cations sit in octahedral and tetrahedral sites. The absorption spectra and the colour of transition metal ions are quite different for these geometries, and so the site occupation can be easily and unambiguously determined. The spinel NiAl2O4 is a case in point. The absorption spectrum of this material reveals that the Ni2 þ ions are found in both positions. Moreover, spectra taken in the earliest stages of the formation reaction:

NiO þ Al2O3 ! NiAl2O4

2 þ show that the Ni occupies both sites from the very start of the reaction. In the related spinel NiGa2O4, the Ni2 þ ions exclusively occupy octahedral sites. Because the structure of glasses cannot be solved by X-ray crystallography it is difficult to obtain structural information at an atomic level, especially concerning the cation coordination in glasses. However, it is often possible to incorporate a small amount of a transition metal into the structure as a probe of local geometry. For example, Figure 7.29 shows a glass bottle incorporating a small quantity of Co2 þ . The blue colour is considered to be typical of tetrahedrally coordinated Co2 þ in oxide matrices and indicates that the medium-sized Co2 þ ions replace the small Si4 þ ions in tetrahedral sites in the glass network. The tetrahedral network structure of silicate glass is well known, but new glasses often have quite unknown structures. As an illustration of the use of transition metal ions as structural probes we can consider an 2 þ exploration of the structure of a ZnCl2 glass. A small amount of Mn incorporated into an (Mn,Zn)Cl2 glass imparts a yellow colour and yields a spectrum typical of tetrahedrally coordinated Mn2 þ . As with the silicate 2 þ glasses, small additions of Co in (Co,Zn)Cl2 gives a blue material characteristic of tetrahedrally coordinated 2 þ Co ions. These results givestrength to the argument that the amorphous ZnCl2 glass is formed from a random 2 þ network of linked ZnCl4 tetrahedral units and that the added ions replace Zn in the network. The suggestion 2 þ is further strengthened by noting that incorporation of Fe in (Fe,Zn)Cl2 glass yields an absorption spectrum expected from tetrahedrally coordinated Fe2 þ ions. This measurement of absorption spectra can also be used to determine the oxidation states of transition metal ions and indirectly to yield information upon the local conditions prevailing during formation reactions. To illustrate this, consider the fabrication of heavy metal fluoride glasses for potential optical fibre use (Section 2.9). In order to gain some insight into the conditions occurring during reaction, a small amount of vanadium was incorporated into a glass composed mainly of ZrF4,BaF2 and NaF. When the glass was made in a nitrogen atmosphere, the colour was yellow green and spectral analysis showed it to contain V3þ in octahedral sites. When a partial pressure of oxygen of about 0.1 atm was introduced into the nitrogen, V5 þ formed, which is colourless and so the glass loses its yellow green hue. Surprisingly, there was no trace found of the stable ion V4 þ under any processing condition. This information allowed the reaction mechanisms occurring within the glass during fabrication to be determined, a task of considerable difficulty by other methods. These examples show that the colour of a transition metal ion and its careful measurement using absorption spectra can give useful structural results in a variety of situations and often over a range of temperatures which remain inaccessible to other experimental techniques. Colour and the Optical Properties of Materials 288

þ Figure 7.29 A bottle coloured blue by the addition of Co2 ions, which occupy tetrahedral sites in the glass matrix

7.15 Colours from Lanthanoid Ions

The lanthanoids (also called the rare earths) have electrons in partly filled 4f orbitals (Appendix A7.1.3). Many lanthanoids show colours due to electron transitions involving the 4f orbitals, and these transitions are similarly forbidden in terms of parity, leading to rather weak coloration (Table 7.11). There is a considerable difference between the lanthanoids and the 3d transition metal ions. The 4f electrons in the lanthanoids are well shielded beneath an outer electron configuration (5s2 5p6 6s2) and so are little influenced by the crystal surroundings. This means that the important optical (and magnetic) properties attributed to the 4f electrons on any particular lanthanoid ion do not depend significantly upon the host structure and the colours do not arise from crystal-field splitting of the f-orbital energies. For this reason the transitions 289 Colour from Atoms and Ions

Table 7.11 Colours characteristic of lanthanoids ions

Ion Electron conﬁguration Characteristic colour þ Ce3 4f1 yellow þ Pr3 4f2 green þ Nd3 4f3 lilac/violet þ Pm3 4f4 pink þ Sm3 4f5 pale yellow þ Sm2 4f6 red/green þ Eu3 4f6 pink þ Eu2 4f7 brown þ Tb3 4f8 pink þ Dy3 4f9 pale yellow þ Dy2 4f10 brown þ Ho3 4f10 yellow þ Er3 4f11 pink þ Tm3 4f12 green

can be usefully labelled with atomic term symbols (Section 7.2). In addition, the transitions from one 4f configuration to another are far narrower than those influenced by crystal-field splitting, as can be judged by 2 þ 2 þ comparison of the absorption spectra of Ni(H2O)6 and Cu(H2O)6 (Figure 7.14) and of ruby (Figure 7.25) with that of Nd3þ ions doped into a sodium oxyfluorideglass (Figure 7.30). (The spectral linewidths emitted by lanthanoid ions in good crystalline matrices are narrower than those in more disordered structures, such as in glasses.) Lanthanoid elements, thus, find use in phosphors (Chapter 9), lasers (Section 7.16) and other light- emitting solids, where a host lattice can be chosen with respect to processing conditions without changing the desirable colour properties of the ion greatly.

1.0

9/2 7/2 3/2 H 2 S G 4 2 + + + 5/2 7/2 F 5/2 4 F G 4 0.5 4

5/2 7/2 D G 2 4 + 9/2 1/2 G 4 P Absorbance (arbitrary units) 2 0 400 500 600 700 800

Wavelength /nm þ Figure 7.30 The main peaks in the absorption spectrum of a glass doped with Nd3 ions. The peaks are labelled with the free-ion terms of the ion. [Data extracted from B. Kartikeyan, S. Mohan, Mater. Res. Bull. 39, 1507–1515 (2004)] Colour and the Optical Properties of Materials 290

The simplest lanthanoid ion is Ce3 þ , with configuration [Xe] 4f1 5d1 6s2, the lowest energy levels arising 2 2 3þ from the single f electron being F7=2 and F5=2. The next higher energy state for Ce is the 5d level. Owing to interaction of the more exposed 5d electrons with the surrounding crystal structure, this is broadened into a band of energies, which also may overlap with another broadened band of energies derived from the 6s energy level (Figure 7.31a). The transitions between the 5d/6s band and the 4f levels are allowed, and the colours produced by transitions of this type are intense. However, the transition does not consist of a sharp line. Instead, any light falling on the crystal will not be absorbed if the energy of the light is less than the energy difference between the 2 2 F7=2 and F5=2 levels and the upper band of energies. When the incident photons have energies greater than this energy gap they will be absorbed. Thus, there will be a sharp change in absorption from low to high (Figure 7.31b). The change in absorption the absorption edge lies towards the violet end of the spectrum. The exact position of the absorption edge is a function of the surrounding matrix and will not be sharp due to lattice vibrations, defects and other factors. In the case of CeO2, which is a good absorber of near-ultraviolet light, the edge of the absorption creeps into the visible. Blue and green are strongly absorbed and to the eye the oxide is perceived as pale yellow. The exact position of the upper energy band is influenced by the surrounding crystal structure. Forming a solid solution with another oxide with a different lattice parameter will move the absorption edge slightly. An example is given by the solid solution formed by reacting yellow CeO2 with Y2O3. The change in structure moves the absorption band further into the ultraviolet and so renders the CeO2 solid solution colourless. These materials transmit the visible spectrum well but strongly absorb ultraviolet and have found use as ultraviolet- absorbing transparent coatings. (Also see Section 10.1.) As the number off electrons increases, the energy-leveldiagrams become increasingly complex (see Chapter 9). However, Eu2 þ , with a configuration 4f7, is an exception. In this case the higher energy state is obtained by transferring an f electron to the outer 5d orbitals, which are lower than all of the other 4f energy levels. As the 5d orbital is exposed to the crystal lattice, this 4f6 5d configuration forms a band of energies (Figure 7.32). As in the case of Ce3þ , transitions from the ground state to the upper energy band are allowed. The energy gap is slightly smaller than in the case of Ce3 þ , and so the absorption moves slightly deeper into the visible spectrum. Because of this, the colour of the oxide EuO is a red brown rather than yellow. The situation in Ce3 þ - and Eu2 þ -containing materials, therefore, is rather similar, which means that these ions are very useful in providing a high-intensity luminescence (see Chapter 9). However, the 5d band is not a continuum of energies, as represented schematically in Figures 7.31 and 7.32, but is usually split into subbands that reflect the crystal-field splitting of the 5d orbitals. Absorption spectra that extend into the ultraviolet thus show a variable number of absorption peaks which can affect emission spectra in this wavelength region.

7.16 The Neodymium (Nd3þ ) Solid-State Laser: A Four-Level Laser

Although the ruby laser (Section 7.11) was the first laser made, three-level operation makes it inefficient, because more than half of the ground state must be excited before a population inversion is possible. A more energy-efficient device can be made employing a four-level energy-level scheme (Figure 7.33). Lasers using this type of energy-level arrangement are referred to as four-level lasers. Laser operation takes place in the following sequence of steps.

1. Atoms in the ground state E0 are excited to a rather high energy level E1 by optical pumping. This process needs to be fast and efﬁcient. 2. Atoms in E1 lose energy again by way of a fast and efﬁcient radiationless process to an intermediate state I1. Once in I1, atoms should have a long lifetime and not lose energy quickly. 291 Colour from Atoms and Ions

eV(a) cm–1

5 40000 Ce3+

35000 5d energy 4 band 30000 Energy 25000 3

20000 ~ 400 nm

2 15000

10000 1

5000

2 F7/2 2 0 F5/2

(b)

40 Absorbance / %

300 400 500 600 700

Wavelength / nm þ Figure 7.31 (a) Schematic energy-level diagram of Ce3 in a typical oxide structure. The emission from the lower edge of the upper energy band to the ground state is close to the far-violet region of the spectrum. The exact position depends upon the host structure. (b) The diffuse absorbance spectrum of CeO2. [Reprinted from J. Solid State Chem, 181, F. Tessier et al., Powder preparation and UV absorption properties of selected compositions in the CeO2–Y2O3 system, 1204–1212, Copyright (2008), with permission from Elsevier] Colour and the Optical Properties of Materials 292

eV cm–1

5 40000 Eu2+

35000 6 4 4f -5d band 30000 Energy 25000 3

20000

2 15000 ~ 420 nm

10000 1

5000

8 0 S7/2 þ Figure 7.32 Schematic energy-level diagram of Eu2 in a typical oxide structure. The emission from the lower edge of the upper energy band to the ground state is in the violet region of the spectrum. The exact position depends upon the host structure

3. It is essential that another intermediate state, I0, is present and also sufﬁciently high above the ground state to be effectively empty. In this case, a small population in I1 gives a population inversion between I1 and I0. 4. Ultimately, a few photons will be released by spontaneous emission as some atoms drop from I1 to I0. These can promote stimulated emission between I1 and I0, allowing laser action to take place. 5. Atoms return from I0 to E0 by a step which needs to be rapid and radiationless. 6. If the energy corresponding to the transitions from E1 to I1 and I0 to E0 can be easily dissipated, continuous operation rather than pulsed operation is possible.

The most important four-levelsolid-state laser uses neodymium (Nd3þ ions) as the activecentres. These ions can be introduced into awidevariety of host lattices with little effect on optical properties because the important 4f orbitals are shielded from the crystal surroundings as described above. The most common host materials are glass, yttrium aluminium garnet (YAG) and calcium tungstate (CaWO4). The important transitions taking place in Nd3þ -ion lasers can be understood in terms of a simpliﬁed energy- level diagram (Figure 7.34). The f-electron levels are rather sharp. Above these lie bands of considerable width derived from the interaction of the 5d and 6s orbitals. Optical pumping excites the ions from the ground state to thesewide bands. This process is very efﬁcient because broad bands allow a wide range of wavelengths to pump the laser and because the transitions are allowed in terms of quantum theory. In addition, loss of energy from the excited state down to the f-electron energy levels is fast. The energy loss halts at the pair of 4F levels. The 293 Colour from Atoms and Ions

Pump Laser emission

E0 Figure 7.33 Schematic arrangement of energy levels and transitions in a four-level laser. Internal (radiationless) transitions are marked with dashed lines

Energy

J eV

8×10–19 5

7×10–19 4 5d-6s band 6×10–19

5×10–19 3

4×10–19

2 3×10–19

4F 2×10–19 1 4 I15/2 –19 1×10 4 I13/2 4 I11/2 4 0 0 I9/2 Figure 7.34 Energy levels of most importance in the neodymium laser. The pump transition is from the ground 4 4 state to the broad 5d–6s band. The main laser transition is between the F and I11/2 levels. Internal radiationless transitions are marked with dashed lines Colour and the Optical Properties of Materials 294

4 4 principal laser transition is from these F levels to I11=2. The emission is at approximately 1060 nm in the infrared. The laser medium contains about 1 % Nd3 þ and can have quite high power outputs. These lasers can be operated continuously or pulsed. At higher Nd3 þ concentrations the lifetime of the 4F upper state drops from about 200 ms in a typically 1 % doped material to about 5 ms at higher dopant concentrations. This is due to Nd Nd interactions and associated changes in lattice vibration characteristics. Under these conditions, laser operation is no longer possible.

7.17 Ampliﬁcation of Optical-Fibre Signals

The amplification of signals in fibre-optic transmission systems is of great importance, as the input signal degrades with distance due to attenuation and dispersion. Originally, amplification used costly repeaters, which transformed the optical pulses into electronic signals, amplified these electronically and then recreated optical pulses. Operating systems are now available which use a section of optical fibre doped with erbium (Er3þ ) as the activator. Erbium-fibre amplifiers using 1.48 and 0.98 mm pump radiation were perfected 1989. The amplifying section consists of about 30 m of monomode fibre core containing just a few hundred parts per million of Er3þ (Figure 7.35a). This section of the fibre is illuminated by a semiconductor diode laser (Section 10.9) at the frequency of the carrier signal. The commonest wavelengths used are 980, 1480 and 1550 nm. The erbium ions transfer energy from the laser to the signal pulses as they traverse this section of fibre. The energy transfer comes about in the following way. Illumination of the erbium-containing section of fibre 4 4 with energy of wavelength 980 nm excites the ions from the ground state ( I15=2) to the upper state ( I11=2) from

(a) Er3+ doped length of fibre

incoming outgoing signal signal

(b) 4 I11/2

4 I13/2

Energy passed

980 nm to signal 1480 nm

4 I15/2 ground state þ Figure 7.35 Signal amplification in an Er3 -doped section of optical fibre. (a) A weak incoming signal is 3 þ substantially amplified on traversing the section. (b) Schematic energy-level diagram of Er in SiO2. Pump 4 wavelengths at approximately 980 and 1480 nm populate the I13/2 level, which passes this energy to the signal 295 Colour from Atoms and Ions

4 whence they rapidly decay to the I13=2 level (Figure 7.35b). This process is referred to as pumping and the laser involved as the pump. The use of radiation of 1480 nm wavelength excites the Er3þ ions directly from the 4 ground state to the I13=2 level.This state has quite a long lifetime. A passing light pulse, with awavelength close to 1480 nm, empties the Er3 þ excited state via stimulated emission (Section 1.9). In effect, the pump energy is transferred to the signal pulses over the course of the erbium-doped stretch of fibre. This achieves signal amplification while retaining the coherence of the pulse constituting the signal. þ The Er3 -doped sections of fibre are made in a similar way to that described in Section 2.9. The gas stream which is used to lay down what will become the core region of the fibre is modified by the addition of erbium chloride and aluminium chloride. The aluminium chloride is added as a co-dopant because it has been found that the presence of Al3þ ions in the glass greatly increases the number of Er3þ ions which can be incorporated before clustering starts to occur. The chlorides decompose in the same way as the chlorides of silicon and germanium, to form soot containing the desired concentration of Er3þ ions. Subsequent heating and collapse of the tube produces a preform with an erbium-doped core. Signal amplification is also used in the National Ignition Facility for fusion research in the USA. This aims to use laser beams to ignite fusion in a deuterium tritium pellet. The output from 192 lasers is used, but as these beams are nowhere near powerful enough, they are repeatedly passed through glass light guides containing Nd3þ inorder to be amplified as described above. Inthis facility, the Nd3þ is pumpedby Xe flash lamps, like the first ruby lasers (Section 7.11), and at each pass the pump energy is added to the laser beam energy until sufficient power has been reached, at which stage all of the beams are focused onto the target (see this chapter’s Further Reading).

7.18 Transition Metal, Lanthanoid and Actinoid Pigments6

Inorganic pigments are colorants used to enhance the appearance of an object. Pigments are incorporated as finely ground powders and are often applied to surfaces as paints and inks. Although organic pigments (Chapter 8) are usually brighter than inorganic pigments, they are not stable at moderate or high temperatures. This poses a major problem in the fabrication of decorative ceramics and glasses, as high temperatures are essential during manufacture (Figure 7.36). The use of transition metal and lanthanoid (or, more rarely, actinoid) compounds, which are added in small quantities to the batch, overcomes this difficulty. The colours generated are often due to the d d or f f transitions described above. The actinoids have partly filled 5f orbitals and behave in a similar way to lanthanoid. These are not usually associated with colour production because of the scarcity and radioactive nature of the heavy atoms. The most commonly utilized examples are uranium compounds, which are used to colour glass and ceramics a yellow green colour. The material used in glasses is usually the yellow trioxide, UO3. Small quantities of UO3 dissolve completely in many glasses to yield a coloured yellow green transparent material (Figure 7.37). It is also used in larger quantities as a green yellow pigment for ceramics. The other commonly utilized uranium compound is uranyl nitrate, UO2(NO3)26H2O, often called uranium nitrate. It forms yellow crystals which are readily soluble in water. Large numbers of inorganic transition metal oxides are used as pigments. Chromic oxide (Cr2O3), known as chrome green, is typical of a number of simple oxides used the green colour arising in the crystal-field splitting of the energy of the d orbitals on octahedrally coordinated Cr3þ ions. Many complex oxides are also used. For example, calcium chromium silicate (Ca3Cr2Si3O12), with the garnet structure, shows a green colour attributed to the octahedrally coordinated Cr3þ ions, but because of differences in the crystal-field splitting of the d-orbital energy levels in the two compounds the colours are perceived as different. Cobalt aluminate

6 The terms ‘lanthanoid’ and ‘actinoid’ are now recommended by IUPAC. See footnote to Appendix A7.1.3. Colour and the Optical Properties of Materials 296

Figure 7.36 An enamel trinket box lid. The enamel is a glass-based material which has been fused to a metal base. The colours are derived from oxide pigments dissolved or dispersed in the glass

2 þ (CoAl2O4), with the spinel structure, is a blue pigment; the colour arises from tetrahedrally coordinated Co ions in the crystal. Similarly, cobalt silicate (Co2SiO4), with the olivine structure, is a blue pigment, which also relies upon tetrahedrally coordinated Co2 þ ions as the colour producer. Differences in the crystal-ﬁeld splitting 2 þ of the Co energy levels in these two crystals give each a unique tone. Cobalt chromite (CoCr2O4), which also

þ Figure 7.37 Yellow–green uranium (U6 )-doped glass 297 Colour from Atoms and Ions adopts the spinel structure, contains tetrahedrally coordinated Co2 þ , imparting a blue colour, and octahedrally 3þ coordinated Cr ions in a similar geometry to that in Cr2O3 to give a green colour. The resulting compound combines both of these tones to yield a blue green pigment. The use of such colorants is hardly new. Several thousand years ago the Egyptians synthesized blue objects using a colorant now known as ‘Egyptian blue’ and the Chinese synthesized both blue and purple artefacts using ‘Han blue’ and ‘Han purple’. The mode of production of Egyptian blue is typical of the techniques used. Artisans heated a mixture of lime, copper oxide and quartz, in approximate ratios of 1:1:4, at high temperatures in a kiln. This produced a polycrystalline/glassy blue solid which was ground to make a blue pigment which could be used in paints. All these ancient blue pigments have been shown to be complex copper silicates. The formulae are CaCuSi4O10 for Egyptian blue, BaCuSi4O10 for Han blue and BaCuSi2O6 for Han purple. The compounds themselves are ring silicates in which the colour is derived from crystal- ﬁeld splitting of the Cu2 þ d-orbitals in a square planar environment. Bearing in mind the fact that the alkaline earth copper silicon oxygen systems are complex and contain a bewildering variety of both coloured and noncoloured crystalline and glassy phases, the technological expertise of the craftsmen was considerable. In the past, the desires for bright colours led to the use of pigments which were dangerous and which would not be allowed today. For example, Scheele’s green, a bright green compound precipitated from ‘arsenious acid’ with copper sulfate solution, which has been assigned an approximate formula HCuAsO3, and Paris green, a mixed copper arsenic acetate of approximate formula 3Cu(AsO2)2Cu(CH3COO)2, nominally copper acetoarsenite, were both widely used to colour much sought after green wallpaper in the nineteenth century. These compounds, however, are rather unstable and release toxic arsenic-containing vapours in moist air. Indeed, the death of Napoleon, on the island of St Helena, in 1821, is attributed to arsenic poisoning arising from the decomposition of Paris green pigments in the wallpapers of his accommodation. In many of these examples, the agent causing the colour is a substituted transition metal or lanthanoid ion. The degree of coloration can be adjusted by changing the amount of dopant or by adjusting co-dopants to change the dimensions of the surrounding crystal structure. Thus, a colourless crystal can be made to appear black by doping with two substituents, one of which absorbs radiation in the low-energy yellow part of the visible spectrum and another that absorbs in the high-energy blue region. The black colour of the pigment CoxZn7 xSb2O12 is attributed to such double absorption. In this material, which adopts the inverse spinel structure, Co2 þ ions occupy both octahedral and tetrahedral sites. The colour is due to the Co2 þ , which þ þ replaces Zn2 to form substitutional defects. The absorption of Co2 in octahedral sites centres on red yellow, and of Co2 þ in tetrahedral sites on blue, giving an overall black material. Despite this long history, there is considerable current research concerned with ceramic pigment formulation.

7.19 Spectral-Hole Formation

The widths of the d d absorption bands in the spectra of transition metal ions in solids are generally considerable, due to the strong interactions of the surrounding crystal matrix with the exposed d orbitals on the cations. In the case of lanthanoid ions, the widths of the f f bands are considerably smaller, but still appreciable due to thermal vibrations. These can be eliminated if the crystal host structure is cooled to 10 K or less. Nevertheless, the narrow lines still have some width. These have been explored both to study fundamental materials properties and for data storage. For this latter purpose, multiple bits are stored at a single location in the host crystal, using the f f transition as the means to this end. The technique is called spectral-hole burning. The method uses crystals doped with lanthanoids, subsequently cooled to close to absolute zero. Under these circumstances, the linewidth of a peak in the spectrum of the ion will, in an ideal case, consist of a single Colour and the Optical Properties of Materials 298

(a)

(b)

(c)

(d)

Figure 7.38 Homogeneous and inhomogeneous absorption of ions in a crystal: (a) homogeneous linewidth of a small subset of ions; (b) inhomogeneous linewidth of the whole set of ions; (c) single spectral hole in the inhomogeneous linewidth; (d) several spectral holes in the inhomogeneous linewidth

excitation frequency called the homogeneous linewidth Gh (Figure 7.38a). However, in most crystals, not all the cation sites are exactly identical. This means that the active centres in the crystal give rise to a narrow spectrum of different absorption frequencies. Of course, the line is still very sharp in spectroscopic terms, but is wider than the single frequency absorption of an individual ion. The total absorption width of a collection of such centres is called the inhomogeneous linewidth Gi (Figure 7.38b). The homogeneous linewidth of typical lanthanoid ions such as Pr3þ ,Sm3þ ,Eu3þ and Er3þ is approximately 10 kHz, while the inhomogeneous linewidth is of the order of 10 GHz, a factor of 106. 299 Colour from Atoms and Ions

conduction band

photo ionisation

f level metastable f level

ground state (a) (b) (c)

Figure 7.39 Mechanisms of spectral hole burning: (a) two level; (b) metastable trapping; (c) gated photo- ionisation

Irradiation of a small volume of crystal with a well-defined beam of laser light of the appropriate frequency will excite only a subset of the atoms that contribute to the homogeneous linewidth. This causes a dip to be recorded in the homogeneous profile at exactly the excitation energy of the subset of ions involved (Figure 7.38c). A change in the irradiation frequency can excite a second subset of cations, and so on. Hence, the homogeneous linewidth becomes pitted (Figure 7.38d). These dips in the profile can be used for storage of several single bits of data at the same location in the crystal the tinyvolume irradiated by the laser. In theory, the number of bits that can be accommodated at the location is equal to the ratio of Gi/Gh. In the case of 3þ one system that has been studied, Er doped into Y2SiO5, Gi ¼ 0.6 GHz and Gh ¼ 50 kHz, so that the number of bits than can be accommodated is 0.6 109/50 103 ¼ 12 000. In order to make a memory store, it is essential that the spectral holes have a reasonable lifetime. In general, a transition between two f f levels, known as two-level hole burning (Figure 7.39a) gives a hole lifetime near to 10 6 s, although hole lifetimes of up to 10 ms have been observed. A memory involving such transitions would need to be refreshed continually, and would not be suitable for long term data storage. The lifetime of a spectral hole can be increased by using different transitions. One method is analogous to the transition from a 4T state to a 2E state in ruby, and involves an intermediate level with a much longer lifetime than two-level hole burning. The mechanism, called metastable trapping (Figure 7.39b), increases the spectral hole lifetime significantly. A third mechanism involves making use of transitions to the broad band of energies that lies above the f f levels. In this, a laser beam initially excites an ion from the ground state to an excited f level. The excited state is then ionised and an electron is promoted into the conduction band of the solid (see Chapter 10) using ‘gated’ photo-ionisation (Figure 7.39c). This results in the creation, for instance, of an Ln3þ ion from a dopant population of Ln2 þ . The process involves irradiation of the crystal volume with the hole-burning wavelength lb and simultaneously with the ‘gating’ wavelength lg. The lifetime of the hole now depends on the rate at which the electron drops from the conduction band to reform the Colour and the Optical Properties of Materials 300

Ln2 þ state. At low temperatures, the lifetime of these spectral holes is considerable and the memory storage is regarded as permanent.

Appendix A7.1 Electron Conﬁgurations

A7.1.1 Electron conﬁgurations of the lighter atoms

Formost chemical purposes anatom or anion can beconsidered toconsist ofadense minutenucleus surrounded by electrons which are said to occupy a series of orbitals. The electron configuration of an atom or an ion describes theway in which the electrons are allocated to these orbitals. The simplest approximation which gives the occupancy of the orbitals is the independent particle model (the orbital approximation), in which each electron is supposedto be isolated andmoving ina field comprising that arising inthe nucleus andall of the other electrons combined. In this approach, each electron is assigned a set of four unique quantum numbers which correspond to the atomic orbital that the electron occupies. The atomic orbitals form a set of shells which are filled from the lowest energy upwards. The Pauli exclusion principle demands that only two electrons, with opposed spins, can occupy an orbital. If this were not so, all electrons would end up in the lowest energy orbital. The lowest energy shell is characterized by a principal quantum number n ¼ 1 and contains only one atomic orbital called an s-orbital. This, like any atomic orbital, can contain either one or two electrons. The two atoms that these two alternatives correspond to are hydrogen (H) and helium (He). The electron configurations of these two atoms are written H1s1 He 1s2 where the principal quantum number (1) is written first, the orbital (s) follows and then the number of electrons in the orbital as a superscript. The next lowest energy shell is characterized by a principal quantum number n ¼ 2 and contains one s-orbital and three p-orbitals,px py and pz, all of which have the same energy. The s-orbital can contain up to two electrons, as above, and the three p-orbitals can contain a maximum of six electrons. The electron configurations of the atoms which make up the second shell, from lowest to highest energy, are Li 1s2 2s1 or [He] 2s1 Be 1s2 2s2 or [He] 2s2 B1s2 2s2 2p1 or [He] 2s2 2p1 C1s2 2s2 2p2 or [He] 2s2 2p2 N1s2 2s2 2p3 or [He] 2s2 2p3 O1s2 2s2 2p4 or [He] 2s2 2p4 F1s2 2s2 2p5 or [He] 2s2 2p5 Ne 1s2 2s2 2p6 or [He] 2s2 2p6

The second shell is now full. Note that in order to write the configuration in a compact form the inner filled shell is represented by the symbol of the atom with that configuration, which is He in this case. The next energy shell is characterized by a principal quantum number n ¼ 3 and contains one s-orbital, three p-orbitals px,py and pz,andfived-orbitals,dxy,dxz,dyz,dx2 y2 and dz2 . The s-orbital can contain up to two electrons and the three p-orbitals can contain a maximum of six electrons, as before. The five d-orbitals can contain up to 10 electrons. Atoms with partly filled d orbitals are called transition metals. The electron configurations of the atoms which make up the third shell, from lowest to highest energy, are: 301 Colour from Atoms and Ions

Na 1s2 2s2 2p6 3s1 or [Ne] 3s1 Mg 1s2 2s2 2p6 3s2 or [Ne] 3s2 Al 1s2 2s2 2p6 3s2 3p1 or [Ne] 3s2 3p1 Si 1s2 2s2 2p6 3s2 3p2 or [Ne] 3s2 3p2 P1s2 2s2 2p6 3s2 3p3 or [Ne] 3s2 3p3 S1s2 2s2 2p6 3s2 3p4 or [Ne] 3s2 3p4 Cl 1s2 2s2 2p6 3s2 3p5 or [Ne] 3s2 3p5 Ar 1s2 2s2 2p6 3s2 3p6 or [Ne] 3s2 3p6

The energy of the 4s-orbital is close to that of the 3d-orbitals and is usually filled before the 3d group. The electron configuration of the 3d transition metals is given in Appendix A7.1.2. The filling of the fourth, fifth and subsequent shells follows along the same lines as above. In heavier atoms there is often some uncertainty in the order in which the orbitals are filled. This will be observed, for example, in some of the atoms listed in Appendix A7.1.3. The completely filled ns2np6 configurations (which correspond to the inert gases), used to write the electron configurations in a compact form, are

He 1s2 Ne [He] 2s2 2p6 Ar [Ne] 3s2 3p6 Kr [Ar] 3d10 4s2 4p6 Xe [Kr] 4d10 5s2 5p6

The outer electron configuration of the atoms is given in Figure 7.2. The electron configuration of ions is written in an identical fashion. Cationic configurations can usually be derived from that of the parent atoms by removing a small number of electrons from the atomic orbitals last filled and anionic configurations by adding electrons to these same orbitals.

A7.1.2 The 3d transition metals

The ten 3d transition metal elements are found in Period 4 of the Periodic Table: K, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Ga, Ge, As, Se, Br, Kr. They are characterized by having partly ﬁlled 3d atomic orbitals (Table A7.1. Cuprous (Cu þ ) and zinc (Zn2 þ ) ions do not behave as typical transition metal ions as they have completely ﬁlled 3d-orbitals.

A7.1.3 The lanthanoid (rare earth) elements

The 15 lanthanoid or rare earth elements7 are found in Period 6 of the Periodic Table: Cs, Ba, (La), Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb,(Lu), Hf, Ta, W... . The electron conﬁguration of some of these atoms is uncertain and neither lanthanum nor lutetium behave as typical lanthanoid, although both are frequently included in the group (Table A7.2).

7 Though ‘lanthanide’ is still in widespread use, the IUPAC recommendation IR 3.5 is that the term ‘lanthanoid’ be used for this group of elements. The reasoning is that ‘ oid’ means ‘having the form of, like, similar to’, whereas ‘ ide’ is normally indicative of negative ions. For the same reason, ‘actinoid’ is now recommended over use of ‘actinide’. Colour and the Optical Properties of Materials 302

Table A7.1 The 3d transition metals

Electron conﬁguration d electron a Name Symbol of atom Ion conﬁguration of ion þ Scandium Sc [Ar] 3d1 4s2 Sc3 d0 þ Titanium Ti [Ar] 3d2 4s2 Ti4 d0 þ Ti3 d1 þ Ti2 d2 þ Vanadium V [Ar] 3d3 4s2 V5 d0 þ V4 d1 þ V3 d2 þ V2 d3 þ Chromium Cr [Ar] 3d5 4s1 Cr3 d3 þ Manganese Mn [Ar] 3d5 4s2 Mn4 d3 þ Mn3 d4 þ Mn2 d5 þ Iron Fe [Ar] 3d6 4s2 Fe3 d5 þ Fe2 d6 þ Cobalt Co [Ar] 3d7 4s2 Co4 d5 þ Co3 d6 þ Co2 d7 þ Nickel Ni [Ar] 3d8 4s2 Ni2 d8 þ Copper Cu [Ar] 3d10 4s1 Cu2 d9 þ Cu d10 þ Zinc Zn [Ar] 3d10 4s2 Zn2 d10 a [Ar] ¼ 1s2 2s2 2p6 3s2 3p6.

Appendix A7.2 Terms and Levels

A7.2.1 The vector model of the atom

The energy levels associated with the electron configurations of an atom (Appendix A7.1) are derived by using the vector model of an atom. In this model, classical ideas are grafted onto the quantum mechanics of the atom. The quantum number l is associated with the angular momentum of the electron around the nucleus. It is represented by an angular momentum vector l. Similarly, the spin quantum number of the electron s is associated with a spin angular momentum vector s. (Vectors in the following text are specified in bold type and quantum numbers in italic type.) The scalar values of s and l are written ms and ml. In the vector model of the atom, the two angular momentum vectors are added together to get a total angular momentum for the atom as a whole. This is then related to the electron energy levels of the atom. There are two main waysoftackling this task. The first ofthese makes the approximation that the electrostatic repulsion between electrons is the most important energy term. In this approximation, called Russell Saunders coupling, all of the individual s vectors of the electrons are summed vectorially to yield a total spin angular momentum vector S. Similarly, all of the individual l vectors for the electrons present are summed vectorially to give a total orbital angular momentum vector L. The vectors S and L can also be summed vectorially to give a total angular momentum vector J. Note that the convention is to use lower case letters for a single electron and upper case for many electrons. One alternative approach to Russell Saunders coupling is to assume that the interaction between the orbital angular momentum and the spin angular momentum is the most important. This interaction is called spin orbit coupling. In this case, the s and l vectors for an individual electron are added vectorially to give a total angular 303 Colour from Atoms and Ions

Table A7.2 The lanthanoid elements

Electron conﬁguration f electron a Name Symbol of atom Ion conﬁguration of ion þ Lanthanum La [Xe] 5d1 6s2 or [Xe] 4f1 6s2 La3 4f0 þ Cerium Ce [Xe] 4f1 5d1 6s2 or [Xe] 4f2 6s2 Ce4 4f0 þ Ce3 4f1 þ Praseodymium Pr [Xe] 4f3 6s2 Pr4 4f1 þ Pr3 4f2 þ Neodymium Nd [Xe] 4f4 6s2 Nd3 4f3 þ Promethium Pm [Xe] 4f5 6s2 Pm3 4f4 þ Samarium Sm [Xe] 4f6 6s2 Sm3 4f5 þ Sm2 4f6 þ Europium Eu [Xe] 4f7 6s2 Eu3 4f6 þ Eu2 4f7 þ Gadolinium Gd [Xe] 4f7 5d1 6s2 Gd3 4f7 þ Terbium Tb [Xe] 4f9 6s2 Tb4 4f7 þ Tb3 4f8 þ Dysprosium Dy [Xe] 4f10 6s2 Dy3 4f9 þ Holmium Ho [Xe] 4f11 6s2 Ho3 4f10 þ Erbium Er [Xe] 4f12 6s2 Er3 4f11 þ Thulium Tm [Xe] 4f13 6s2 Tm3 4f12 þ Ytterbium Yb [Xe] 4f14 6s2 Yb3 4f13 þ Yb2 4f14 þ Lutetium Lu [Xe] 4f14 5d1 6s2 Lu3 4f14

a [Xe] ¼ 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6. momentum vector j for a single electron. These values of j are then added vectorially to give the total angular momentum vector J, for the whole atom. The technique of adding j values to obtain energy levels is called j j coupling. Broadly speaking, Russell Saunders coupling works well for lighter atoms and j j coupling for heavier atoms. Other coupling schemes have also been worked out, and these ﬁnd use in medium and heavy atoms. In reality, the energy levels derived from each scheme represent approximations to those found by experiment. For almost all purposes, the Russell Saunders coupling scheme is adequate for the speciﬁcation of the energy levels of an isolated many-electron atom. In general, it is not necessary to work directly with the vectors S, L and J. Instead, many-electron quantum numbers (not vectors) S, L and J are used to label the energy levels in a simpleway. The method of derivation is set out in Appendix A7.2.2. Thevalue of S is not used directly, but is replaced by the spin multiplicity, 2S þ 1. Similarly, the total angular momentum quantum number L is replaced by a letter symbol similar to that used for the single electron quantum number l (Table A7.3). After L ¼ 3, F, the sequence of letters is alphabetic, omitting J. Be aware that the symbol ‘S’ has two interpretations: S (roman) is the value of L and S (italic) as the value of total spin. The combinations are written in the following form:

2S þ 1L

This is called a term symbol. It represents a set of energy levels, called a term in spectroscopic parlance. States with a multiplicity of one are called singlet states, states with a multiplicity of two are called doublet states, with Colour and the Optical Properties of Materials 304

Table A7.3 The correspondence of L values and letter symbols

L Symbol 0S 1P 2D 3F 4G 5H multiplicity of three, triplets, with multiplicity four, quartets and so on. Hence, 1S is called singlet S and 3Pis called triplet P. A7.2.2 Energy levels and terms of many-electron atoms

The Russell Saunders terms of an atom are derived by adding the individual spin quantum numbers of the electrons to yield a total spin quantum number S and adding the individual orbital angular momentum quantum numbers of the electrons to give a total orbital angular momentum quantum number L. For example, the total spin angular momentum quantum number S(2) for two electrons is given by adding the individual quantum numbers thus:

Sð2Þ¼ðs1 þ s2Þ; ðs1 þ s21Þ; ...; js1s2j:

1 As s1 and s2 are both equal to 2:

Sð2Þ¼1or0

(Note that this is a maximum number of values. If two electrons are spin paired then only the value zero applies. If the electrons are in different orbitals, say an s and a p orbital, then they can have parallel or antiparallel spins, making both one and zero possible.) In order to obtain the value of S for three electrons S(3), the value for two electrons S(2) is added to the spin quantum number of the third electron s3 thus:

Sð3Þ¼ðSð2Þþs3Þ; ðSð2Þþs31Þ; ...; jSð2Þs3j

Both of the values for S(2) are permitted, so we obtain:

ð Þ¼ ; ð Þ¼ þ 1 ; þ 1 ¼ 3 ; 1 S 2 1 S 3 1 2 1 2 1 2 2

ð Þ¼ ; ð Þ¼ þ 1 S 2 0 S 3 0 2

ð Þ¼3 ; 1 S 3 2 2

This procedure is called the Clebsch Gordon rule. It is used to obtain the S values for increasing numbers of spins. It will be found that for an even number of electrons, S values are integers, and for an odd number of electrons, S values are half-integers. 305 Colour from Atoms and Ions

The total angular momentum quantum number L is obtained in a similar fashion. For two electrons with individual angular momentum quantum numbers l1 and l2, the total angular momentum quantum number L(2) is:

Lð2Þ¼ðl1 þ l2Þ; ðl1 þ l2 1Þ; ...; jl1 l2j In the case of three electrons, the Clebsch Gordon rule is applied thus:

Lð3Þ¼ðLð2Þþl3Þ; ðLð2Þþl3 1Þ; ...; jLð2Þl3j using every value of L(2) obtained previously. Values of the quantum number L are given letter symbols as described. For example, the terms arising from the two p electrons on carbon (C), with l1 ¼ l2 ¼ 1, are obtained in the following way:

¼ 1 þ 1 ; 1 1 ¼ ; S 2 2 2 2 1 0

2S þ 1 ¼ 3or1

L ¼ 1 þ 1; 1 þ 11; 11 ¼ 2; 1; 0 ðD; P; SÞ

The total number of possible terms for the two p electrons is given by combining these values. The possible terms for two p electrons are therefore:

3D; 3P; 3S; 1D; 1P; 1S

Not all of these possibilities are allowed for any particular conﬁguration, because the Pauli exclusion principle limits the number of electrons in each orbital to two with opposed spins. When this is taken into account, the allowed terms are:

3P; 1D; 1S

Similarly, an atom with two d electrons, with a conﬁguration of, say, 3d2, will again have:

¼ 1 þ 1 ; 1 1 ¼ ; S 2 2 2 2 1 0

The possible values of L are obtained by using the values l1 ¼ l2 ¼ 2, to give:

L ¼ 2 þ 2; ð2 þ 21Þ; ð2 þ 22Þ; ð2 þ 23Þ; ð2 þ 24Þ¼4; 3; 2; 1; 0 ðG; F; D; P; SÞ

Combining all of these gives:

3G; 3F; 3D; 3P; 3S; 1G; 1F; 1D; 1P; 1S

Taking into account forbidden conﬁgurations gives the allowed terms as:

3F; 3P; 1G; 1D; 1S Colour and the Optical Properties of Materials 306

The energies of the terms are difﬁcult to obtain simply, and they must be calculated using quantum mechanical procedures. However, the lowest energy term, the ground-state term, is easily found using the method described in Appendix A7.2.3.

A7.2.3 The ground-state term of an atom

The lowest energy term, the ground-state term, can be found using Hund’s ﬁrst and second rules:

1. The term with the lowest energy has the highest multiplicity, equivalent to the highest total spin quantum number S. 2. For terms with the same value of multiplicity, the term with the highest value of L is lowest in energy.

There is a simple method of determining the ground state of any atom or ion. The procedure is as follows:

1. Draw a set of boxes corresponding to the number of orbitals available. For a p electron, this is three (Figure A7.1). 2. Label each box with the value of ml, highest on the left and lowest on the right. 3. Fill the boxes with unpaired electrons, from left to right. When each box contains one electron, start again at the left. þ 1 1 4. Sum the ms values of each electron, 2 or 2. This is equal to the maximum value of S. 5. Sum the ml values of each electron to give a maximum value of L. 6. Write the ground term 2S þ 1L. Using this technique, set out in Figure A7.1, the ground term of the 2p2 and 2p4 conﬁgurations is 3P.

ml 1 0 –1

2 S = ½ + ½ = 1 p 2S+1 = 3 L = 1 + 0 = 1

term scheme 3P

ml 1 0 –1

4 S = ½ + ½ + ½ – ½ = 1 p 2S+1 = 3 L = 1 + 0 + –1 + 1= 1

term scheme 3P Figure A7.1 Determination of a ground state term

A7.2.4 Energy levels of many-electron atoms

The term symbol does not account for the true complexity found in most atoms. This arises from the interaction between the spin and the orbitalmomentum (spin orbit coupling) that is ignored in Russell Saunders coupling. For this the quantum number J described above is needed. It is given by:

J ¼ðL þ SÞ; ðL þ S1Þ; ...; jL Sj 307 Colour from Atoms and Ions where |L S| is the modulus (absolute value, regardless of whether þ or ). Thus, the term 3P, has J values given by:

J ¼ð1 þ 1Þ; ð1 þ 11Þ; ...; j11j¼2; 1; 0

2S þ 1 The new quantum number is incorporated as a subscript to the term, now written Lj and this is no longer called a term symbol, but a level. Each value of J represents a different energy level. It is found that a singlet term always givesone energy level, a doublet two, a triplet three and so on. Thus, ground-state term 3P is composed of 3 3 3 three levels P0, P1 and P2. The separation of these energy levels is controlled by the magnitude of the interaction between L and S. Hund’s third rule (see Appendix A7.2.3 for rules 1 and 2) allows the values of J to be sorted in order of energy. The level with the lowest energy is that with lowest J value if the valence shell is up to half full and that with the highest J value if the valence shell is more than half full. The nomenclature just described is not adequate to describe either molecular energy levels or the energy levels of atoms in crystal ﬁelds. In these cases a terminology based upon symmetry is most often encountered.

Further Reading

Background reading on the molecular orbital theory of molecular energy levels of molecules, together with a description of the associated spectroscopies, is given by P. W. Atkins, Physical Chemistry, 5th edition, Oxford University Press, 1994 (especially Chapters 16 and 17). P.W. Atkins, J. de Paula, R. Friedman, Quanta, Matter and Change, Oxford University Press, 2008 (especially Chapters 10 and 11). Fireworks are described in M. S. Russell, The Chemistry of Fireworks, Royal Society of Chemistry, Cambridge, 2000. The colour of water is discussed by C. F. Bohren, Clouds in a Glass of Beer, Dover, New York, 2001, Chapter 20 (originally published by John Wiley and Sons, Inc., New York, 1987.

The colours of water and ice and of liquid and solid D2O are discussed by T. Quickenden and A. Hanlon, Chem. Br. 36 (12), 37 39 (2000), the references cited therein and the subsequent correspondence: Chem. Br. 37 (2), 19 (2001); 37 (3), 18 (2001).

Sonoluminescene can be tracked starting from H. Xu, N. G. Glumac, K. S. Suslick, Angew. Chem. Int. Ed. 49, 1079 1082 (2010). An introduction to organic chemistry is found in J. McMurry, Organic Chemistry, 6th edition, Brooks/Cole, Belmont, CA, 2004. Information on the structures and colours of organic molecules will be found in P. F. Gordon, P. Gregory, Organic Chemistry in Colour, Springer-Verlag, Berlin, 1983. J. Griffiths, Colour and Constitution of Organic Molecules, Academic Press, London, 1976. P. Rys, H. Zollinger, Fundamentals of the Chemistry and Applications of Dyes, John Wiley and Sons, Inc., New York, 1972. Information on many aspects of the materials in this chapter is given by R. M. Christie, Colour Chemistry, Royal Society of Chemistry, Cambridge, 2001. P. Bamfield, Chromic Phenomena, Royal Society of Chemistry, Cambridge, 2001. A vast amount of information on porphyrins is contained in K. M. Kadish, K. M. Smith, R. Guilard (eds), The Porphyrin Handbook, Vols 11 19, Academic Press, San Diego, 2003. Of relevance to this chapter, see Vol. 19, Applications of Phthalocyanines. An extensive description of he colours found in plants is given by D. Lee, Nature’s Palette, The Science of Plant Colour, University of Chicago Press, Chicago, IL, 2007. T. Bechtold, R. Mussak (eds), Handbook of Natural Colorants,JohnWileyandSons,Ltd,Chichester, 2009. The following review, and the references cited therein, gives much information on flower colours: K. Yoshida, M. Mori, T. Condo, Blue flower colour development by anthocyanins: from chemical structure to cell physiology. Nat. Prod. Rep. 26, 884 915 (2009). The history of the discovery of mauve is given by S. Garfield, Mauve, Faber and Faber, London, 2000. Colour and the Optical Properties of Materials 362

For information on blueprints, see M. Ware, Cyanotype: The History, Science and Art of Photographic Printing in Prussian Blue, Science Museum, London, 1999. Pigments, from the point of view of an artist, are the subject of V. Findlay, Colour; Travels through the Paintbox, Folio, London, 2009. Details of analysis of metal ions using colour tests is given in F. Feigl, V. Anger, Spot Tests in Inorganic Analysis, Elsevier, Amsterdam, 1971. Earlier editions of this volume, author F. Feigl alone, are equally useful Colorimetric sensor arrays are described by K. S. Suslick, Mater. Res. Soc. Bull. 29, 720 725 (2004). An excellent introduction to photochromic materials is given by H. G. Heller, Photochromics for the future, in Electronic Materials, from Silicon to Organics, L. S. Miller, J. B. Mullin (eds), Plenum, New York, 1991, p. 471. For further information, see J. Crano, R. J. Guglielmetti (eds), Organic Photochromic and Thermochromic Compounds, New York, Plenum, 1999. 9 Luminescence

. How do ﬂuorescent tubes produce light? . How do plasma displays produce colours? . How do glow sticks produce colours?

This chapter is concerned with a number of aspects of colour that are of commercial and social importance. In day-to-day life, the most important of these may well be fluorescent lighting, which is widespread in homes and offices, providing a relatively low-energy method of illumination. Fluorescence microscopy and the use of green fluorescent protein have had an impact upon medicine and health which cannot be overstated. Both of these are described, together with some other aspects of luminescence which are of related interest. Here, it is useful to note that the term luminescence essentially means ‘light production’. It does not imply that a single process operates in all cases. Thus, the mechanism of light emission by a firefly is quite different from light emission by a fluorescent bulb, although both may be termed luminescence.

9.1 Luminescence

The emission of light by bodies at relatively low temperatures, ‘cold light’, is generally called luminescence, which can be contrasted with light emission by a hot body, called incandescence (Section 1.6). Solids that give rise to luminescence are called phosphors or, latterly, luminescent materials. Investigations into luminescence have a long history. The term phosphor is derived from the element phosphorus. This element was ﬁrst isolated in about 1674 or 1675 by the alchemist Brandt, who discovered that the material shonewith a pale greenish light in the dark and gave it the name phosphorus, which is from the Greek phos (light) and phero (I carry). The name

Colour and the Optical Properties of Materials Richard J. D. Tilley 2011 John Wiley & Sons, Ltd Colour and the Optical Properties of Materials 364 for the glow from phosphorus, phosphorescence, was taken over and quite widely applied to many other forms of ‘cold light’, including that from decaying organisms. The modern story of luminescence can be thought to start with Ritter, in 1801, who, investigating solids that would glow after being illuminated with daylight, discovered that the effect was greatest when the sample was placed in the dark region beyond the violet end of the spectrum. He postulated, therefore, that an invisible form of ‘light’, termed ultraviolet, existed (Figure 9.1).

Figure 9.1 (a) A pale yellow phosphor based on zinc sulﬁde (ZnS) in normal daylight. (b) The same material irradiated with ultraviolet light with a wavelength range of 350–380 nm, showing a bright yellow–green ﬂuorescence 365 Luminescence

In 1852 Stokes published the results of an extensive investigation in which he invented the term fluorescence for the light that he observed emerging from crystals of the mineral fluorspar after illumination. Stokes’ law, proposed at this time, stated that the radiation emitted has a longer wavelength (lower energy) than the exciting radiation. The wavelength difference is known as the Stokes shift. A.-E. Becquerel, in France, also studying the emission of light by solids after illumination, was of the opinion that fluorescence was simply a short-duration form of phosphorescence in which he was essentially correct. Stokes’ fluorescence is characterised by the by the immediate release of the exciting energy as light, while A.-E. Becquerel’s phosphorescence is typified by the slow conversion of the exciting energy into light, so that light emission is delayed by a length of time that can vary from milliseconds to hours or days. Now it is appreciated that the two expressions represent extremes on a continuum that can be defined in terms of a quantum mechanical probability of the emission of visible radiation. Roughly coincident with the fluorescence and phosphorescence studies of Stokes and Becquerel, researches by Faraday, Giessler, Crookes and others showed that gases and some solids produced light when bombarded with ‘cathode rays’ (Section 7.5). To distinguish this light from fluorescence and phosphorescence the effect was called cathodoluminescence. Yetanother form ofluminescencehad an enormousimpact upon twentiethcenturyscience. H. Becquerel, the son of A.-E. Becquerel, was studying phosphorescence when he discovered, in 1896, that uranium salts emitted a radiation, ‘uranium rays’, that were new. Pierre and Marie Curie followed this up and discovered radioactivity and the highly radioactive element radium. The radioactivity is intense and this causes many materials to emit light, now called radioluminescence.1 Radium was widely used in luminous paints for watch and other instrument dials that could be seen in the dark. This has now ceased due to the harmful effects of the radiation emitted. These preceding studies pointed the way towards a general understanding of luminescence. It is now clear that luminescent materials are able to gain energy from an energetic, often ‘invisible’ source (ultraviolet light, electric fields, X-rays, energetic particles from radioactive decay, and so on2) and re-emit some of this energy in the form of light. For this reason, luminescence is now subdivided into a number of categories depending upon the nature of the exciting source (Table 9.1). Recall that there are no general mechanisms for luminescence, and apart from fluorescence and phosphorescence, which are two aspects of the same process, each type needs to be treated independently. Phosphors are widely used in, for example, fluorescent lamps (ultraviolet to visible), old-fashioned cathode ray (CR) tube TV (electron impact to visible) and scintillators (X-rays, g-rays and energetic subatomic particle impact to visible). Molecular fluorescence is of increasing importance in the study of living organisms and medical sciences via fluorescence microscopy and related techniques.

9.2 Activators, Sensitisers and Fluorophores

The ﬁrst commercial phosphor, ‘Balmain’s paint’, calcium sulﬁde (CaS), was produced in 1880. Partly as a result of the desire to make better commercial materials, it was discovered that in many instances pure compounds would not show luminescence, although the same material when contaminated with minute traces of impurities was luminescent. Moreover, the colour of the luminescence was dependent upon the chemical nature of the impurity.

1 Marie Curie described the phenomenon thus: ‘One of our joys was to go into our workroom at night; we then perceived on all sides the feebly luminous silhouettes of the bottles or capsules containing our products. It was really a lovely sight and one always new to us. The glowing tubes looked like faint, fairy lights.’ 2 Materials that emit light in response to the impact of high energy particles or X rays are often termed scintillators rather than phosphors. Colour and the Optical Properties of Materials 366

Table 9.1 Types of luminescence

Type Deﬁnition Source of energy Fluorescence Electronic decay between allowed states Ultraviolet and visible photons Phosphorescence Electronic decay between forbidden states Ultraviolet and visible photons Bioluminescence Luminescence in a living organism Gibbs energy of chemical reactions Cathodoluminescence Luminescence due to electron Electron kinetic energy bombardment (cathode ‘rays’) Chemiluminescence Luminescence during a chemical reaction Gibbs energy of chemical reaction Electroluminescence Luminescence resulting from the Electrical potential energy application of an electric ﬁeld Photoluminescence Luminescence after irradiation by visible or Ultraviolet and visible photons ultraviolet light Radioluminescence Luminescence as a result of radioactivity Energetic particles and g rays Tenebrescence Reversible darkening under irradiation Photon or particle energy Thermoluminescence Luminescence following an increase of Thermal energy temperature Triboluminescence Luminescence following fracture Chemical bond energy or friction

This can be illustrated with respect to the fluorescence of minerals when irradiated with ultraviolet light. (No distinction will be made between fluorescence and phosphorescence here.) A few examples are: white or agate (SiO2), aragonite (CaCO3), calcite (CaCO3), gypsum (CaSO42H2O), fluorite bluish white: (CaF2), halite (NaCl), wollastonite (CaSiO3); red: barite (BaSO4), calcite (CaCO3), corundum (Al2O3), halite (NaCl), sphalerite (ZnS); orange: barite (BaSO4), calcite (CaCO3), scheelite (CaWO4), sphalerite (ZnS), wurtzite (ZnS), zircon (ZrSiO4); yellow: agate (SiO2), calcite (CaCO3), diopside (CaMgSi2O6), scheelite (CaWO4), talc (Mg3Si4O10(OH)2), wollastonite (CaSiO3), zincite (ZnO), zircon (ZrSiO4); green: agate (SiO2), aragonite (CaCO3), calcite (CaCO3), opal (SiO2), willemite (Zn2SiO4); blue: albite (NaAlSi3O8), calcite (CaCO3), fluorite (CaF2), gypsum (CaSO42H2O), sphalerite (ZnS), wollastonite (CaSiO3).

The first point of note is that many minerals appear frequently and show different fluorescent colours. This indicates that the crystal matrix is simply acting as a (nominally inactive) host that has a small quantity of impurity or activator (A), incorporated within it. The role of the host structure or of the host activator combination is to absorb an excitation in the form of a photon of energy hn1. The activator re-emits the excitation as a photon of energy hn2. The colour emitted is dependent upon the nature of the activator. Sometimes it is found that the activator-containing material cannot absorb the exciting radiation directly, in which case a helper species, a sensitiser, is needed as well. In this case the sensitiser absorbs the exciting photons, of energy hn3, and passes the energy to the activator. The sensitiser can be the crystal matrix itself or a specially introduced centre such as another cation (Figure 9.2a). Irrespective of whether the luminescence is derived from an activator sensitiser pair, or just from the activator alone, a rapid decay of light is characteristic of fluorescence. On the other hand, a slow decay is characteristic of phosphorescence (Figure 9.2b). In this case, the energy is often regarded as being stored in 367 Luminescence

h 2 out (a)

A heat S h in 1 h 3 in heat

energy transfer

(b)

phosphorescence Luminous intensity fluorescence

Time

Figure 9.2 Schematic representation of energy absorption and emission processes taking place in a luminescent material. (a) Absorption of radiation. A represents an activator centre and S a sensitizer centre. The photons n n n absorbed and emitted, h 1,h 2 and h 3, need not necessarily all be different. Some energy is also lost to the host structure as vibrational energy (heat). (b) Emission of radiation. Fluorescence is characterised by a rapid decay of intensity, while phosphorescence is characterized by a slow decay a reservoir from which it slowly leaks. This feature is more commonly associated with heavy atoms, and is one of the reasons why H. Becquerel was interested in uranium compounds. Although a large amount of study and research has focused on inorganic phosphors, because of applications in lighting, TV tubes and displays, the fluorescence of organic molecules is equally important. In these systems, the activator is usually the molecule itself, and such molecules are said to show autofluorescence. In larger molecules, only a specific group of atoms might be involved in the fluorescence, and, by analogy with the term chromophore, the group is labelled a fluorophore. In this sense, a fluorophore in an organic molecule is the equivalent of an activator in an inorganic phosphor. However, this term is used rather imprecisely, and often it is applied to any small fluorescent molecule. Thus, the molecule fluorescein (see below) is often called a fluorophore. As with inorganic materials, organic fluorescent organic compounds may need a sensitiser. This may just be a different part of the same large molecule or be the surrounding matrix or solvent. Colour and the Optical Properties of Materials 368

(a) H H N C N CC N C N H H H H

O O NaO2 S SO2 Na

(b) Na+- O O ONa-+

Figure 9.3 The structures of fluorescent molecules: (a) C. I. fluorescent brightening agent 30, used in detergents; (b) soluble fluorescein, used in protective clothing and ophthalmic medicine

Fluorescence from organic molecules has long been used by the manufacturers of detergents used for washing white clothes. A colourless dye is incorporated into the detergent which fluoresces blue when irradiated with ultraviolet light. The small amount of ultraviolet in natural daylight is sufficient to create this effect, but illumination by ultraviolet light in, for example, a club can make these garments almost shine. The structures of these fluorescent brighteners generally consist of linked benzene rings, together with groups to aid in the solubility in water and incorporation into the cloth (Figure 9.3a). Similar additives can be found in cosmetics which are designed to glow when illuminated by the ultraviolet light in discos and nightclubs. Fluorescein (C20H12O5, Figure 9.3b), a fluorescent compound, is one of a remarkable family of coloured materials closely related to phenolphthalein (Figure 8.28). Fluorescein is a yellow red powder with an intense green fluorescence. Fluorescein itself is rather insoluble and is more often met with as ‘soluble fluorescein’, which is the disodium salt, Na2C10H10O5, which is freely soluble in water. The excitation radiation maximum is close to 495 nm (blue green) and the fluorescence wavelength is 519 nm (green). The effect of the absorption and fluorescence is to impart an unmistakable intense yellow green fluorescence to solutions. It is widely used to colour safety garments and is the familiar yellow green marker colour used to highlight passages of text. It is also the bright yellow green colour that is used in eye examinations and contact lens fitting.

9.3 Atomic Processes in Photoluminescence

There are two basic atomic processes that must take place during photoluminescence: (i) photon absorption; (ii) photon emission. In addition, energy transfer between excited and nonexcited states is often important and, indeed, vital when sensitisers are a necessary component of the luminescent system. Some of these processes are listed in Table 9.2 and are discussed at various points throughout this chapter.

9.3.1 Energy absorption and emission

The initial process that takes place in ﬂuorescence is the absorption of a photon of the exciting radiation, E1 ¼ hn1. (No distinction will be made between ﬂuorescence and phosphorescence here.) For simplicity, 369 Luminescence

Table 9.2 Photoluminescence processes

a Process Example Lifetime/s Absorption of photons Ground state absorption (GSA) A þ hn ! A 10 16 10 15 Excited state absorption (ESA) A þ hn ! A Multiphoton absorption A þ nhn ! A

Emission of photons Fluorescence (spontaneous emission) A ! A þ hn þ phonons (allowed transition) 10 12 10 6 Phosphorescence (spontaneous emission) A ! A þ hn þ phonons (forbidden transition) 10 6 1 Stimulated emission A þ hn ! A þ 2hn

Photon conversion þ nhn ! ! þ hn n n Up conversion (UC) A 1 A A 2 ( 1 < 2) þ hn ! ! þ nhn n > n Quantum cutting (QC) A 1 A A 2 ( 1 2)

Energy distribution and quenching Molecular collision A þ Q ! A þ Q þ phonons Defect A þ De ! A þ De þ phonons Internal conversion (IC) A ! A þ phonons Intersystem crossing (ISC) 1A* ! 3A*1012 10 4 Energy transfer (ET) S þ A ! A þ S; A þ A ! A þ A; A þ Q ! A þ Q Cross relaxation (CR) A þ A ! A þ A a A: activator, luminescent centre ground state; S: sensitiser ground state; Q: molecule; De: surface or bulk defect; : excited state; : doubly excited state; phonons are equivalent to heat energy. Note that intersystem crossing can also involve other multiplicities and is not confined to singlet–triplet pairs. The lifetime gives an approximation to the length of time that the process takes. assume that the absorption takes place at the activator, which is excited from the normal low-energy ground state A to an excited state A. The activator subsequently emits the fluorescent photon and returns to the ground state. Because the emitted radiation is at a greater wavelength (lower energy) than that absorbed, some energy DE1 is redistributed from light energy into another form. Schematically, the activator drops down through a closely spaced set of energy levels to a new lower state (Figure 9.4a). The transitions responsible for this degradation are generally called nonradiative transitions, decay or relaxation.The energy deficit DE1 generally ends up within the phosphor matrix, in the form of lattice vibrations or phonons, i.e. heat. The activator then returns to the initial state, emitting a photon as it does so, in a radiative transition conforming to E2 ¼ hn2. The lower energy level reached may not be the ground state, but one of several higher states associated with the lower level. Radiationless transitions will again disperse this extra energy DE2 as vibrational energy in the host matrix until the centre reaches the final state of lowest energy (Figure 9.4b). Note that these nonradiative transitions are often drawn as if they occur between vibrational energy levels. This is not mandatory, and transitions can be between electronic energy levels as long as the 4 2 energy can be carried away by phonons successfully. The T2g ! E transition in ruby is an example (Sections 7.10 and 7.11). The difference in energy between the exciting radiation and the emitted radiation is:

DE ¼ DE1 þ DE2 Colour and the Optical Properties of Materials 370

excited state A* (a) (b) ΔE1

ν ν E = h 1 E = h 2

ΔE2 ground state A

Stokes shift

(c)

excitation

emission Intensity

λ1 λ2 = hc /E1 = hc /E2 Wavelength λ

Figure 9.4 Energy transfer in phosphors (schematic): (a) absorption by activator; (b) luminescent photon emission; (c) the Stokes shift between the wavelength of the excitation and the emission pulses

The wavelength equivalent of this energy difference:

hc hc Dl ¼ ð9:1Þ E1 E2 constitutes the Stokes shift (Figure 9.4c). Under intense irradiation, an excited state A can absorb a second photon to reach a higher energy state A, a process called excited-state absorption. The excited state can then undergo similar vibrational losses before return to the ground state (Section 9.9). Similarly, a ground state A can absorb several low-energy photons simultaneously to jump to the excited state A (Section 9.11.4).

9.3.2 Kinetic factors

The difference between ﬂuorescence and phosphorescence can usefully be discussed in terms of the kinetics of allowed and disallowed transitions. For light atoms, organic molecules, proteins and so on the instantaneous production of light (ﬂuorescence) can be regarded as due to a spin-allowed transition (DS ¼ 0). The delayed production of light (phosphorescence) is attributed to a spin-forbidden transition (DS ¼ 1, 2, etc.). Frequently, 371 Luminescence

excited state 1 A* excited state 3 A* ISC

fluorescence ν E = h 3 ν E = hν E = h 1 2 phosphorescence

ground state 1 A Figure 9.5 Intersystem crossing (ISC) in which a fluorescent molecule changes from an excited state with spin multiplicity 1 to an excited state with spin multiplicity 3, leading to phosphorescence molecular phosphorescence is associated with the transformation of a singlet excited state 1A* into a triplet excited state 3A* that takesplacemorerapidly than the downwardfluorescencetransition. These changes can be displayed on energy-level diagrams called Jablonski diagrams, which set out the electronic and vibrational energy levels of a molecule in a schematic way, with singlet and triplet states shown in separate columns. Phosphorescence arises when a molecule in a singlet excited state is transformed into a triplet state (Figure 9.5). In this process, called intersystem crossing, the vibrational energy levels of both states coincide and the molecule can transform from one multiplicity to the other without requiring energy input. In the triplet state, radiationless transfer of energy continues until the molecule lies at the lowest energy level of the triplet state. Further emission by a photon is slow because of the selection rules prohibition. In the case of heavy atoms such as transition metals and lanthanoids, mixing of the wave functions on the atoms leads to spin orbit coupling and in reality the spin states are not as well defined as the multiplicity symbol 4 ! 4 D ¼ suggests. Thus, although the T2 A2 transition in ruby can be labelled fluorescence ( S 0) and the 2 ! 4 D ¼ E A2 transition as phosphorescence ( S 2), in practice transitions are often allowed or disallowed to a varying degree by virtue of both the spin and parity selection rules.

9.3.3 Quantum yield and reaction rates

The quantum yield, which measures the efﬁciency of the ﬂuorescence, is given by:

N FðlÞ¼ pe ð9:2Þ Npa

where Npe is number of photons emitted during fluorescence and Npa is the number of photons of the exciting radiation of wavelength l absorbed. The quantum yield reflects the number of ways that the excited state can lose energy. If every excited centre loses energy by only one reaction, rapid photon emission, then the quantum yield will be unity. Quantum yields of 10 % or more may be satisfactory for some applications, but much higher quantum yields are always desirable and are mandatory for some specialist devices. Because the numbers of photons emitted and absorbed are measured over a certain time span, the quantum yield is a measure of the rate of fluorescence, which can be treated in terms of chemical kinetics. The rate of decay of the excited state A will simply be given by the sum of the rates of all the deactivation reactions that contribute to the loss of energy of the excited state, including fluorescence. Colour and the Optical Properties of Materials 372

The simplest case to consider is when the rate of decay of the exciting centres is simply a function of the number of excited centres that are formed; in chemical terms, the concentration of A, written [A]. The rate of the reaction is given by the differential form:

d½A* ¼k½A* ð9:3Þ dt where d[A]/dt is the rate of decay of the excited state and k is the rate constant of the process. This dependence is termed ﬁrst order and in this case the rate of decay follows ﬁrst-order kinetics:

½ ¼½ kt ¼½ t=t ð : Þ A* t A* 0e A* 0e 9 4

where [A ]t is the concentration of excited centres at time t after a pulse of excitation radiation has generated an initial population of [A ]0, t is the elapsed time and t is the fluorescence lifetime. The fluorescence lifetime is, then, the time taken for the number of excited centres to decay to a value of 1/e of the initial value at t ¼ 0. The luminescence lifetime and phosphorescence lifetime are defined in the same way. Note, though, that the difference between fluorescence and phosphorescence is simply a matter of rate of reaction and there is no value of t that arbitrarily separates one from the other. The number of excited centres is assessed by measuring the radiant exitance, and this can be substituted for [A] in these equations. Note that in almost all literature this is termed ‘intensity’, given in arbitrary units, and is plotted in the form:

t=t It ¼ I0e

A plot of the (natural) logarithm of the radiant exitance (or It) emitted against time will give a straight line of slope k ¼ 1/t (Figure 9.6). Departure of the plotted curve from an exponential form is evidence that the mechanism of light emission is more complex than that supposed. If a luminescent material loses energy by first-order processes due to fluorescence and phosphorescence, both will depend upon [A] in the way described by Equations 9.3 and 9.4. The rate constant k of the overall reaction is now given by the sum of the rate constants for fluorescence and phosphorescence, kF and kP:

k ¼ kF þ kP

The quantum yield for fluorescence FF(l) is given by the number of fluorescent photons emitted in a certain time compared with the total number absorbed and then used up in the two competing processes of fluorescence and phosphorescence. It is then possible to write Equation 9.2 in terms of the rate constants as:

kF FFðlÞ¼ kF þ kP

Similarly, the quantum yield for phosphorescence FP(l) is given by the number of phosphorescent photons emitted in a certain time compared with the total number absorbed and then used up in the two competing processes of ﬂuorescence and phosphorescence, leading to:

kP FPðlÞ¼ kF þ kP 373 Luminescence

(a) 1.0

0.5 Radiant exitance / arbitrary units 0 0.5 1.0 1.5 2.0 2.5 3.0 Time / arbitrary units

(b)

slope = –1/τ Ln (Radiant exitance)

Time / arbitrary units Figure 9.6 First-order kinetics of ﬂuorescence: (a) exponential decay of radiant exitance with time (schematic); (b) the slope of a plot of ln(radiant exitance) versus time gives the reciprocal ﬂuorescence lifetime 1/t

It follows that if the excited centre loses energy by a number of other ﬁrst-order processes that compete with ﬂuorescence, the overall reaction rate constant is given by:

k ¼ kF þ kP þ kX þ kY þ ð9:5Þ and the ﬂuorescence quantum yield is given by:

kF FFðlÞ¼ ð9:6Þ kF þ kP þ kX þ kY þ

A number of luminescent materials exhibit light production, often at a very low level, for much longer than the lifetime indicates. This is called afterglow, and is distinct from phosphorescence. Afterglow is, in broad Colour and the Optical Properties of Materials 374 terms, due to electrons becoming trapped at a sitewhere they are prohibited from losing energy. It is prevalent in semiconductors (Chapter 10), where both trapped electrons and trapped holes can cause afterglow. Generally, the electrons (or holes) are released from trapping sites by thermal energy, after which the normal luminescent process can take place. Afterglow is a problem for some applications.

9.3.4 Structural interactions

The role of the surrounding medium is often important in processes that lead to luminescence. For example, the width of the absorption and emission peaks is verydependent upon the interaction of the orbitals on the activator with the surrounding matrix. In the case of transitions that take place between well-shielded inner orbitals, such as f f transitions, the importance of the external structure is masked, and fluorescence emission lines are narrow. However, if the orbitals involved interact with the outer matrix, such as d d transitions (Cr3 þ ) or d f transitions (Eu2 þ ) or p s transitions (Sb3 þ ), the fluorescent emission bands are wide. These differences are of considerable importance when the performance of fluorescent lamps and other fluorescence-based displays is evaluated. Solvatochromism, the change in colour of a material due to a change in solvent polarity, provides another example of the way in which the surroundings influence fluorescence. The change in colour is described as negative solvatochromism if the colour shift moves the emission to shorter wavelengths (a hypsochromic or blue shift) as the polarity of the solvent increases. It is called positive solvatochromism if the colour shift moves the emission to longer wavelengths (a bathochromic or red shift) as the polarity of the solvent increases. The colour shift comes about because the ground-state energy, the excited-state energy, or both are modified by the surrounding solvent. Solvatochromism is then a manifestation of a change in the position of the electronic absorption and emission bands from a fluorophore. It is often displayed by polar molecules, i.e. molecules with an observable dipole moment. The energy of the ground statewill be influenced by the interaction of the solvent with the dipole on the molecule. If the solvent is nonpolar, typically a hydrocarbon solvent, little interaction will occur. If the solvent is polar, such as water or an alcohol, the interaction may be large. In such a molecule, the excitation of an electron from the ground-state orbital to the excited-state orbital will significantly alter the dipole on the molecule. The interaction with the solvent will then be different in the excited state to that in the ground state. Thus, a change in the polarity of the medium containing the fluorophore will alter the relative positions of the excited- and ground-state energy levels. Adding these effects together gives rise to the overall change in colour. Charge-transfer colours that are associated with cation-to-ligand or ligand-to-ligand electron transfer are also susceptible to solvatochromic effects. The orbitals on the ligands aregenerally exposed to the surroundings and this has an effect upon the ground-state energy. Transfer of an electron from a cation to a ligand, or from one ligand to another, creates an excited state, the energy of which is also influenced by the surroundings. The net difference between the two will then vary if the surroundings change. Although solvatochromism was originally described in terms of molecules in solution, the definition now includes colour change due to the influence of any external surroundings, including a solid matrix.

9.3.5 Quenching

In many circumstances the ability of a normally luminescent centre to emit fluorescence is suppressed or inhibited. This feature is called fluorescence quenching. Quenching is said to be dynamic if the inhibition involves the excited state and static if it involves the ground state in such a way as to prevent the excited state from forming. Quenching is not the result of just a single process, but can be caused by a multiplicity of reactions that are able to compete with the fluorescence mechanism. Several are described below. 375 Luminescence

9.3.5.1 Thermal quenching In solid phosphors, thermal quenching, the reduction or suppression of luminescence due to increase in temperature, is of importance in many applications (Figure 9.7a). Some lamps, for example, can become quite hot during operation. Thermal quenching will then drastically reduce the amount of light given out. The reason for thermal quenching lies in the vibrations of the surrounding matrix. At very low temperatures these are minimal. Electronic excitation will promote a luminescent centre from a low vibrational level in the ground state to a low vibrational levelin the excited state (Figure 9.4a). A considerablegap between the upper and lower energy levels is present that is bridged by the emission of a photon. As the temperature increases, higher and higher vibrational energies are occupied in the ground state and excited state. Ultimately, the ground-state and

10 (a)

CaWO4 3+ Eu :Gd2O3 5 Relative exitance

400 500 600 700 800 900 Temperature / K

excited state (b)

ν E = h 1

ground state

2 3 þ Figure 9.7 Thermal quenching: (a) relative exitance emitted by CaWO4 (WO4 fluorophore) and Eu doped 3 þ into Gd2O3 (Eu activator); (b) schematic depiction of the vibrational energy levels of the ground and excited states at high temperatures. The excited centre can move into the ground state entirely via nonradiative steps Colour and the Optical Properties of Materials 376 excited-state vibrations appear as if merged and the potential energies of both the ground and excited states are virtually the same. In this case the differentiation between the two configurations is blurred and it becomes possible for the excited state to pass from an excited-state vibrational energy level to a ground-state vibrational energy level without photon emission (Figure 9.7b). In molecular terms, the multiplicity of the two states is the same and the transfer of the system from the excited electronic state to upper echelons of the ground state is called internal conversion (IC). Further transitions down the ground-state vibrational energy-level ladder return the system to the ground state solely via nonradiative phonon transitions. IC is a first-order reaction obeying the kinetics given by Equations 9.3 and 9.4. The rate constant kIC can be incorporated into Equations 9.5 and 9.6.

9.3.5.2 Energy transfer Energy transfer away from the excited centre to another centre or the surroundings will quench the fluorescence. This is a process akin to IC, in that the energy transfer is radiationless. The best understood mechanism of energy transfer is Forster€ resonance energy transfer (FRET), sometimes called fluorescence resonance energy transfer. The energy absorbed by the fluorescent centre in the reaction:

A þ hn ! A* is given by:

DEA ¼ hnA

The frequency nA is called the resonant frequency for the transition. If the resonant frequency matches a similar frequency on a nearby quencher molecule Q (the resonance condition), i.e.:

DEA ¼ hnA ¼ DEQ ¼ hnQ then energy can be transferred from A to Q thus:

A* þ Q ! A þ Q*

The centre which provides the energy, A, is often called the donor, and the centre which receives the energy, Q, is often called the acceptor. In addition to the resonance condition, energy transfer can only take place if a suitable interaction is present between the two centres. This can be the overlap of suitable wavefunctions, electric or magnetic dipole interactions or, more rarely, other multipole interactions. This latter condition implies that resonant energy transfer will only occur when the two centres are very close (Figure 9.8a). These conditions can be summarized graphically in terms of the overlap of the absorption spectrum of Q (the acceptor) and the emission spectrum of A (the donor) for the transition in question. The rate of energy transfer is proportional to the area of overlap between the two spectra (Figure 9.8b). In general, energy transfer will be in competition with other processes, such as ﬂuorescence. The relative rates of all of these processes will then contribute to the effectiveness of the energy transfer mechanism. For the sake of simplicity, assume that the only two processes that occur are either ﬂuorescence from A or energy transfer to Q. The critical separation R0 of the centres, the Forster€ distance, can be taken as that at which the rates of these two processes are equal; that is:

kF ¼ kET 377 Luminescence

(a) ET A* Q*

Δ ν hν F E = h A = Q

A Q R

(b) A* emission spectrum Q absorption spectrum Intensity

overlap

Wavelength Figure 9.8 Forster€ resonance energy transfer (FRET). (a) Two competing processes, energy transfer (ET) and ﬂuorescence (F), are possible when the energy-level separation of the A and A and the S and S centres are equal. (b) The efﬁciency of the energy exchange is proportional to the degree of overlap of the emission spectrum of A and the absorption spectrum of Q

If the distance R between the centres is greater than R0, then ﬂuorescence from A occurs, whereas energy transfer is preferred if R is less than R0. For many systems the efﬁciency of energy transfer ZFRET is given by:

6 Z R0 FRET 6 þ 6 R0 R

The value of R0 is of the order of 2 5 nm.

9.3.5.3 Concentration quenching Concentration quenching occurs in many inorganic phosphors. In this phenomenon, the fluorescent centres show good quantum yields when present at low concentrations. However, when the concentration of the luminescent centres increases beyond a certainvalue, whether because of distortions of the surrounding crystal structure, because of clustering or simply because the luminescent centres are close enough for their electron orbitals to interact, excitation energy can be passed to an adjoining centre and so does not result in emission of a photon. There are a number of mechanisms for this energy transfer, which depend upon the closeness of the centres and the way in which they interact. The two main routes are known as direct energy transfer (ET) and cross- relaxation (CR). Energy transfer may allow energy migration through the structure by jumping from one centre Colour and the Optical Properties of Materials 378 to another. Energy transport in this way is well known in crystals containing a high concentration of Gd3 þ ions, where energy is efficiently transferred through the Gd3 þ sublattice jumping from one ion to another. Although energy transferred in this way does not mean that photon emission cannot eventually occur, quantum efficiency is frequently impaired if significant energy transfer occurs. Concentration effects are not confined to the luminous centres alone. Impurities or defects in the solid, especially those near to surfaces, can accept energy. If these reach critical concentrations, then luminescence is throttled. For this reason, much effort is directed towards improving the crystalline perfection of phosphor powders. Concentration quenching is often observed in solutions of fluorescent molecules. The concentration effect can involve only the active molecule, in which case the effect is also called self-quenching, or it may involve an added quenching molecule called a quencher. Fluorescein is a self-quenching molecule and anthracene (C14H10) is quenched by indole 2,3-benzopyrrole (C8H7N) molecules. In solution, concentration quenching is frequently modelled in terms of the collisions between the fluorescent species and the quenching species. The simplest form that this can take is a bimolecular reaction:

A* þ Q ! A þ Q where Q is a molecular quencher. The rate of such a reaction is given by:

Rate ¼ kM½A*½Q

where kM is the rate constant of the reaction, [A ] is the concentration of the ﬂuorescent molecules and [Q] is the concentration of the quencher molecules. In the case of dynamic quenching, the quantum yield is given by the Stern Volmer equation:

F ðlÞ 0 ¼ þ ½ FðlÞ 1 KSV Q

where F0(l) is the quantum yield in the absence of a quenching molecule, F(l) is the quantum yield when the concentration of quenching molecules is [Q] and KSV is the Stern Volmer constant. The value of the Stern Volmer constant is given by:

KSV ¼ t0kQ where t0 is the ﬂuorescence lifetime of the ﬂuorescent species in the absence of the quenching molecules and kQ is the rate constant of the quenching reaction due to molecular species Q. The Stern Volmer equation is often written in terms of the radiant exitance, M0/M, or the lifetimes, t0/t:

M t 0 ¼ 0 ¼ 1 þ K ½¼Q 1 þ t k ½Q M t SV 0 Q where M0 is the radiant exitance in the absence of the quencher, M is the radiant exitance in the presence of the quencher, t0 is the ﬂuorescence lifetime in the absence of the quenching molecules and t is the ﬂuorescence lifetime inthe presence of the quencher. A plot of F0/F, M0/Mor t0/t will yield alineargraph with a slope of KSV (Figure 9.9). 379 Luminescence

Φ0 / Φ

slope = KSV

Concentration [Q] F F Figure 9.9 Idealised Stern–Volmer plot. 0 is the quantum yield in the absence of a quenching molecule, is the quantum yield when the concentration of quenching molecules is [Q] and KSV, the Stern–Volmer constant, is given by the slope of the graph

9.4 Fluorescent Lamps

Fluorescent lamps utilize photoluminescence for light generation. Fluorescent lighting for advertising was first used in 1925, and development of phosphors during the 1930s led to the commercial introduction of low-voltage fluorescent lamps in 1939. The intensity of the luminescence is roughly proportional to the amount of phosphor that is exposed to exciting radiation. Early phosphors were not especially efficient, and the first fluorescent lamps were in the form of tubes about 1 m in length. Improvements in phosphor specification have made the efficiency greater, and since the 1980s compact fluorescent lighting has become commonplace. These lamps, used for indoor lighting, contain an inert gas and a small quantity of mercury vapour at a low pressure. Under electron bombardment from the current passing through the lamp the Hg atoms are excited and emit copious ultraviolet radiation. This consists mainly of line emissions with wavelengths 185, 254 and 365 nm, as well as some radiation in the visible (Section 7.7). Conversion of the ultraviolet radiation to visible is by way of a phosphor coated onto the inside of the tube (Figure 9.10).

9.4.1 Halophosphate lamps

3 þ These lamps use modiﬁed calcium ﬂuorophosphate (Ca5(PO4)3F) as the host matrix. When doped with Sb 3 þ 2 1 1 ions as activator (written as Ca5(PO4)3F:Sb), a blue emission is produced. The Sb ions absorbvia an s to s p transition centred at 254 nm, which closely matches the mercury vapour output. A minor problem with Sb3 þ is that the blue emission gives the lamps a rather cool colour. If Mn2 þ is also incorporated into the system as a coactivator then a warmer tone is produced, as this ion produces an orange red emission (Figure 9.11). Variation in the proportions of Sb to Mn varies the tone of the light. Note that the emission bands are very broad because the orbitals involved in the electron transitions producing the light output interact strongly with the surrounding crystal matrix. There are a number of other aspects of the phosphor which are of interest. First, although the Mn2 þ has good emission characteristics, it is found to be unsatisfactory when used alone. Fortunately, the Sb3 þ acts as a sensitiser for the Mn2 þ ion, thus avoiding another component in the phosphor. Second, the Mn2 þ and Sb3 þ ions occupy the Ca2 þ positions in the host matrix. Now, while Mn2 þ incorporation will not pose an þ þ electroneutrality problem, as the Mn2 ions have the same charge as the Ca2 ions that they replace, this is not so with Sb3 þ . The introduction of Sb3 þ ions into the phosphate will thus cause an internal charge Colour and the Optical Properties of Materials 380

(a)

visible light (b)

glass tube

phosphor e– Hg

ultraviolet

vis ble light

Figure 9.10 Fluorescent lamps: (a) schematic ﬂuorescent tube lamp; (b) processes occurring in the lamp. Electrons (e ) from the cathode collide with mercury (Hg) atoms, which emit ultraviolet radiation which is converted into visible light by a phosphor coating on the inside of the glass tube imbalancewhich will result ina degradation of performance. Toovercome this, charge balance is maintained by adding one F or Cl ion to the phosphate for each Sb3 þ ion. It has been found that an empirically derived composition for the host matrix of Ca10P6F1.8Cl0.2O24 is most satisfactory. These lamps are still available and work continues on improving their performance.

orange-red (Mn2+)

blue (Sb3+) Emission (arbitrary units)

400 500 600 700 Wavelength / nm þ þ Figure 9.11 Emission spectra from Sb3 (blue emission) and Mn2 (orange–red emission) in a typical halophosphate ﬂuorescent tube phosphor 381 Luminescence

9.4.2 Trichromatic lamps

Over the years, improvements have occurred in fluorescent lighting, especially with respect to the colour of the light produced. Trichromatic (Colour 80) lamps produce a very good spectral balance by using a phosphor mixture which emits equal amounts of the colours red, blue and green. The commonest fluorescent centres used are lanthanoid or transition metal ions. Lanthanoid ions have a set of unfilled 4f orbitals. Electron interactions give rise to a large number of electronic energy levels. The 4f orbitals are shielded from the surrounding matrix by outer 5p, 5d and 6s orbitals which contain electrons, so that the 4f energy levels are sharp and similar to those in isolated atoms or ions. The orbitals, 4d, 5p and 6s, all interact strongly with the environment and instead of presenting sharp energy levels they are broadened into wider band of energy (see Chapter 10 for more information on this). Moreover, transitions from a 4f energy level to one of these orbitals are quantum mechanically allowed. This combination of factors means that excitation can be achieved by awide range of exciting wavelengths. Energy loss via photon emission usually takes place between f energy levels. These transitions are forbidden for the same reasons as transitions between the 3d orbitals discussed earlier, but spin orbit coupling and the admixture of other orbital states means that f f transitions are available (Section 7.15). 3 þ The favoured red emitter in trichromatic lamps is Eu doped into a Y2O3 matrix, Y2O3:Eu, with the 3 þ 3 þ 3 þ 6 7 Eu ions occupying the Y sites. The ground state of the Eu 4f ion is F0. The broad band at higher energy is due to a charge transfer transition in which an oxygen 2p electron is transferred to the Eu3 þ to makea4f7 configuration (Figure 9.12a). This charge transfer band absorbs efficiently at 254 nm and accordingly readily takes up the ultraviolet radiation given off by the excited mercury atoms. Subsequent nonradiative decay allow the ion to end up in one of the 5D levels, from which a return to theground state is 5 ! 7 by photon emission. The main transition is between the energy levels D0 F2, leading to emission at a wavelength near to 611 nm. The green emission is from Tb3 þ . This ion absorbs the mercury emission poorly and is coupled with a sensitiser, usually Ce3 þ , which is able to absorb the 254 nm wavelength mercury radiation efficiently due to a charge-transfer band between the oxygen 2p orbitals and the Ce3 þ 5d orbitals. This absorbed energy is then transferredtotheTb3 þ ions. The green emission is at wavelength close to 540 nm, mainly from a

eV cm–1 O 2p - 4f band 4f7 - 5d band 5 40000

4f6 - 5d band 4 30000

5 5 D D3 3 3 5 D2 5 Energy 20000 5D D4 5D 1 2 0

10000 1 7 7 F0 F6 7 7 8 0 F0 F6 S7/2 Eu3+ (4f6) Tb3+ (4f8) Eu2+ (4f 7 ) þ þ þ þ Figure 9.12 Schematic partial energy-level diagrams for the luminescent ions Eu3 ,Tb3 and Eu2 . The Tb3 þ ions do not absorb ultraviolet radiation but gain it by energy transfer from a Ce3 sensitiser Colour and the Optical Properties of Materials 382

green (Ce3+, Tb3+)

orange-red (Eu3+)

Emission (arbitrary units) blue (Eu2+)

400 500 600 700 Wavelength / nm Figure 9.13 The emission spectrum of a trichromatic ﬂuorescent lamp (schematic)

5 7 5 7 5 7 D4-- F5 transition (Figure 9.12b). Three other peaks of lesser intensity occur: D4-- F6, 489 nm; D4-- F4, 5 7 589 nm; D4-- F3, 623 nm. Host matrices are La(Ce)PO4,LaMg(Ce)Al11O19 and La(Ce)MgB5O10. In each case the Tb3 þ and Ce3 þ ions replace La3 þ ions and no charge compensation is needed. The blue emission is produced by Eu2 þ ions, which have a 4f7 electron configuration. This leads to a 8 particularly simple energy-level diagram where the ground state is S7=2 and the upper energy band corresponds to the transfer of an electron into the outer 5d orbital to give a configuration 4f6 5d and ensuring that ultraviolet radiation is absorbed efficiently (Figure 9.12c). The d orbitals interact with the surrounding anions and the exact position of the band depends upon the host crystal. Thus, the luminescent colour of the Eu2 þ centre will be modified by changing the site in the host lattice and the type of host structure. The emission spectrum of the usual tricolour lamp phosphor, BaMgAl10O17:Eu, has a maximum at 450 nm. An emission spectrum from a trichromatic fluorescent tube is illustrated in Figure 9.13. The emission lines are narrow compared with those in Figure 9.11 because the f orbitals involved in the process are shielded from the surrounding matrix. As with the fluorophosphates lamps, the overall emission colour can be modified by changing the relativeamounts of the three phosphors present so as to emphasise the red, green orblue ends of the spectrum.

9.4.3 Other ﬂuorescent lamps

The colour spectrums of the ﬂuorescent lamps described above, although satisfactory for manypurposes, do not give an accurate impression of the colour of an object compared with that perceived when the same object is viewed in daylight. Toovercome this, deluxe (Colour 90) lamps can be used. These employ modiﬁed phosphors so that the emissions are shifted slightly and a fourth phosphor is added to the blend. This latter phosphor, 3 þ 2 þ Y3Al5O12 doped with Ce , absorbs some of the blue violet light emitted by Eu and emits yellow light in its 383 Luminescence

5 40000 5d band 4 Ð1 30000

3 Energy / eV 20000 Energy / cm 2

10000 1

2 F7/2 2 0 F5/2 Ce3+ (4f1)

þ Figure 9.14 Schematic partial energy-level diagram for Ce3 . The transitions shown correspond to the absorption of blue light of wavelength approximately 466 nm and emission at approximately 537 nm. A charge þ transfer band from O 2p to Ce3 5f that absorbs strongly at 256 nm is not shown place. The Ce3 þ absorption is from the 4f1 ground state into the 5d orbitals (Figure 9.14). These higher energy orbitals interact strongly with the environment surrounding the ion and are split due to the crystal-ﬁeld interaction. The two lowest absorption bands are at 342 and 460 nm, and it is this latter transition that is important for absorption of the 450 nm blue emission from Eu2 þ . Following absorption, luminescence is from 2 2 the lower edge of the d band to the ground state, F5=2 and the close F7=2 level (Figure 9.14). These transitions give an output luminescence with a wavelength maximum close to 565 nm. Mercury lamps for street lighting use a high pressure of mercury vapour and produce an emission that is more or less continuous between the limits of 250 and 550 nm (Section 7.7). This output is unbalanced from a visual viewpoint and it is desirable to introduce an ultraviolet-absorbing phosphor that will emit in the red, so as to balance the output. A favoured phosphor for this purpose is a mixed strontium magnesium phosphate using tin 2 þ as an activator, (Sr,Mg)3(PO4)2:Sn , which emits at 630 nm. However, this phosphor is not ideal, and many other materials are currently being explored. Suntanning beds also make use of phosphors, but in this case the main output is required to be in the ultraviolet. UVA, wavelength range 320 400 nm, and UVB, wavelength range 280 320 nm, are both used for this purpose. However, as health concerns over the relationship between ultraviolet irradiation and skin cancer have surfaced, phosphors have been modiﬁed to alter the ratios of these components. Initially, sun bed tubes 2 þ used SrMgSi2O7:Pb, in which Pb is the activator. These gave a broad emission centred on 350 nm and spanning both UVAand UVB. Unfortunately, this materialhas alow stability and was later replaced by BaSi2O5 with Pb2 þ activator, which gave a narrower band centred at 350 nm, limited to UVA.Today, sun beds often use 3 þ tubes containing a mixture of BaSi2O5:Pb and SrAl12O19 containing Ce activator, which has an emission peak centred at approximately 310 nm, thus providing some UVB output.

9.5 Plasma Displays

Plasma displays are, in essence, gas-discharge lamps and the working principle of these displays mirrors that of ﬂuorescent lamps. Monochromatic plasma displays were ﬁrst used in some portable computers in about 1988 (see Figure 7.8). These used an ionised gas to produce an orange red colour in a similar fashion to that exploited Colour and the Optical Properties of Materials 384

Figure 9.15 Plasma screen, circa 2009: (a) side view showing the slim profile compared with CRT displays; (b) detail of display image showing colour rendition by neon signs (Section 7.5). The gas was confined in a series of wells, and two grids of transparent electrodes, one running horizontally and one vertically, provided the necessary current and voltage to ionise the gas. These monochromatic displays rapidly gave way to full-colour displays, which are now commonplace (Figure 9.15). Currently (2010), full-colour plasma display screens dominate the large-screen market. They have an advantage over other display technologies in that the light emitted does not vary greatly with viewing angle. In addition, plasma display television sets arewidely available and at the moment appear to have the edge in providing high-definition television over competing technologies. A display panel consists of a pair of glass plates containing a series of cells each of which acts as a small fluorescent lamp. To form the lamp array, the region between the glass sheets is divided up into sub-pixels by a series of ribs or separators controlled by two sets of electrodes arranged at right angles to each other (Figure 9.16a). Each lamp is several hundred micrometres in size, and there are several million such lamps in a display. Each pixel consists of three lamps, giving off red, blue and green light, making the luminance and resolution uniform across the display that does not vary with viewing angle. The working gas in the cells is a mixture of helium and xenon. When a high voltage is applied across the two electrodes above and below a well the gas is excited into a state resembling a plasma; that is, an electrically neutral mixture of electrons, positive and negative ions. The excited gases emit ultraviolet radiation. As with all the inert gases, the energy-level diagrams are complex and numerous wavelengths are emitted. The principal wavelengths, though, are at 147 nm from excited Xe and at 172 nm from an Xe excimer.3 Each well is coated internally with a red, green or blue phosphor (Figure 9.16b). The layer of magnesium oxide (MgO) serves as a dielectric to enhance the electric field present in the well. The ultraviolet light excites the phosphors to emit in the red, green or blue, in the same way that the ultraviolet light from a mercury vapour excites the phosphors in a fluorescent tube.

3 An excimer is an electronically excited pair of atoms that are not bonded in the ground state. 385 Luminescence

(a) glass

phosphor (r, g, b) glass

(b) glass

insulating layer electrode magnesium oxide

neon + xenon separator

phosphor

electrode

glass

Figure 9.16 (a) The layer structure of a full-colour plasma display. Each well in the structure contains a red, green or blue phosphor. (b) Detail of one subpixel. A mixture of neon and xenon emits ultraviolet light which interacts with the phosphor to give out either red, green or blue light

3 þ The main phosphors used at present are yttrium gadolinium borate doped with europium ((Y,Gd)BO3:Eu ) 2 þ for red emission, barium magnesium aluminate doped with europium (BaMgAl14O23:Eu ) for blue emission 2 þ and zinc silicate doped with manganese (ZnSiO4:Mn ) for green light. None of these is ideal in every way. For example, the red emitter is rather too orange in hue, the blue emitter degrades rather too rapidly under the intense ultraviolet irradiation and the green emitter has a long decay time, which can lead to image blurring. Research on improving these phosphors is intense. In addition, the use of phosphors that utilise quantum cutting (Section 9.10), so that more than one visible photon is emitted per ultraviolet photon absorbed, is highly attractive from the point of view of improving luminosity whilst lowering damage and the other drawbacks mentioned.

9.6 Cathodoluminescence and Cathode Ray Tubes

9.6.1 Cathode rays

Cathodoluminescence is light emission due to irradiation with electrons. This effect was discovered during early researches on the effect of electric ﬁelds on gases at low pressures (Section 7.5). A pair of electrodes were sealed into an evacuated glass tube and subjected to a high voltage. Electrons are expelled from the cathode and are subsequently accelerated towards the anode by the applied voltage. A hole in the anode allowed these ‘cathode rays’ to pass through and hit the glass (or later a phosphor coating), which then gave out light. The Colour and the Optical Properties of Materials 386

deflection anode + plates phosphor coated screen

cathode - cathode rays

evacuated glass tube

Figure 9.17 CRT (schematic). The cathode is a pointed hairpin filament and the anode consists of a cylinder, allowing the ‘cathode rays’ to pass through. The anode and cathode are usually contained in an assembly called an electron gun. There can be several sets of magnetic or electrostatic deflection plates after the gun, allowing the cathode ray spot to be displaced so as to display information process was called cathodoluminescence, and the evacuated tube assembly as a whole was, and still is, called a cathode ray tube, often abbreviated to CRT (Figure 9.17). The first CRT device, invented by Braun in 1897, was the oscilloscope. These instruments display voltage variation with time. The voltage to be displayed is applied to a set of deflection plates or magnetic coils within the tube. These displace the electron beam by an amount proportional to thevoltage. A beam scanned across the screen of the CRT will then not follow a straight line, but display a wave-form, mimicking the signal voltage. Radar (an acronym for radio detection and ranging) was a development of CRT technology that took place in the 1930s and 1940s. In a radar detector, the screen is circular and the linear display of an oscilloscope is now rotated in the plane of the screen every few seconds or so. An antenna sends out a pulsed radio-frequency signal which is reflected by various objects and hence returned to the antenna, which also acts as a receiver, or to a specialized receiving antenna. A disturbance on the rotating signal line, bright spots or breaks, indicates the reception of a radar signal. The position on the line gives the distance to the disturbance and the orientation of the line at that moment gives the direction. Initially used for military applications, such as the detection of enemy aircraft or shipping, problems arose because the signal can also be reflected by storms, rain, snow and ice crystals in the atmosphere. These early ‘problems’ have been exploited and are now the basis of weather radar in common use for meteorological purposes.

9.6.2 Television tubes

The mass market for CRTs was the development of television at the end of the 1940s and on. Towards the end of the century, the market expanded when television-like CRTs became used as computer terminal displays. The electrons are produced by heating a metal filament, which forms the cathode of the device, in an evacuated glass envelope. They are accelerated towards a perforated cylindrical anode by the application of a high voltage. The anode and cathode assembly is often referred to as an electron gun. The far end of the tube assembly is flattened and phosphor coated to form the screen upon which the light-emitting image is formed. The electrons emerge from the anode as a narrow collimated beam which is scanned horizontally and vertically across and down the screen in a predetermined raster pattern by electrostatic or electromagnetic means. The beam is made to cover the screen in a fraction of a second. As the focused beam sweeps past a dot of phosphor, light is emitted. More than a million phosphor dots of approximately 300 mm diameter for each of three primary colours are deposited on the curved screen in routine manufacturing operations. The screen, therefore, is lit up by small spots of phosphor whose emission is refreshed at each pass of the electron beam. Light emission is controlled by variation of the electron flux as the beam scans across the screen. 387 Luminescence

Figure 9.18 The arrangement of red (r), green (g) and blue (b) phosphor dots on a colour cathode-ray television tube.

In both monochrome and colour televisions the light emitted is perceived by the eye in the same way that impressionist pointillist paintings are, by way of additive coloration. For black-and-white displays, two phosphors are used: a blue emitter and a yellow emitter. Colour television utilises three primary colours, arranged in an array (Figure 9.18). Just as in pointillist paintings, it is important that the phosphor spots do not overlap in colour displays, otherwise the picture quality is degraded. Apart from this, there are a number of important parameters that have to be closely controlled to give a good picture. The efficiency of a fluorescent material depends upon many factors, including whether an activator alone is needed or whether a sensitiser and an activator are involved. The concentrations of the activator and sensitiser play an important part in controlling efficiency, and optimal concentrations have to be found experimentally. Impurities generally have a negative effect, and so high purity is a necessity. In addition, the decay time of the phosphor must be suitable. The exitance (invariably termed the intensity) of the light given out by a phosphor after excitation is removed is frequently given by an equation of the type:

It ¼ I0 expðt=tÞ where It is the exitance after time t has elapsed, I0 is the initial exitance the moment excitation ceases and t is the decay time (Section 9.3). The decay time is the time taken for the luminescent radiation to decay to 1/e of its initial value. Clearly, for a continuous picture to be observed, the decay time must be longer than the time that it takes the electron spot to complete its round trip and refresh the emission again. However, this must not be so long that shadow images persist after the action has moved on. In this respect, afterglow can be troublesome. Afterglow is said to occur when the luminescence decays at the rate expected until a certain value is reached and then decays much more slowly. This is frequently due to the presence of impurities in the material which contribute differing mechanisms to the emission process. In summary, a phosphor must have (a) a high efficiency, (b) a suitable decay time, (c) a suitable emission spectrum and (d) a low afterglow level. Thesewere achieved some decades ago and the technology of television tube manufacture is mature from this point of view. Black-and-white TVuses silver-activated zinc sulfide (ZnS:Ag þ ) to give a blue colour plus silver-activated zinc cadmium sulfide ((Zn,Cd)S:Ag þ ) to give a yellow. The ratio of these two phosphors controls the overall hue of the screen. The blue colour produced by ZnS:Ag þ is due to an electron transition between defects in the ZnS structure that are deliberately introduced by silver doping. The nominal reaction to make the phosphor is between þ 2 þ 4 0 silver sulfide (Ag2S) and ZnS, to produce silver (Ag ) substituted for zinc on Zn sites (AgZn ) and

4 Defect notation follows that of Kroger and Vink (see this chapter’s Further Reading). Colour and the Optical Properties of Materials 388

(a) (b) (c)

conduction band

• • V 2 V 2 S 2• S VS

blue yellow blue

Ag ′ ′ Zn ′ AgZn AgZn

valence band

(d) (e) (f)

• • Cls AlZn • Cls

blue blue green

′ ′ ′ CuZn AgZn AgZn

Figure 9.19 Schematic diagram of the defect energy levels in ZnS and (Zn,Cd)S phosphors used in television CRTs: (a) B&W, blue; (b) B&W yellow; (c) colour, blue; (d) colour, blue; (e) colour, blue; (f) colour, green

2. sulfur vacancies (VS ): ð Þ! 0 þ þ 2. Ag2S ZnS 2AgZn SS VS 2. The transition giving rise to light emission is due to the transfer of an electron trapped on a sulfur vacancy VS 0 to a silver ion AgZn (Figure 9.19a). (In semiconductor terminology (see Chapter 10), the sulfur vacancy forms a donor level just below the valence band and the silver ion forms an acceptor level just above the conduction band in the ZnS band gap, and the transition is from a donor level to an acceptor level.) The colour-producing electrons arise in the following way. To a first approximation, the valence band is fully occupied by electrons and the conduction band is completely empty and the defects introduce energy levels into the energy gap between the valence and conduction bands. Irradiation by cathode rays promotes electrons from the valence band into the conduction band. This is an extremely energetic process. Whereas one ultraviolet photon will promote one electron from a lower energy level into the conduction band, a cathode ray electron might promote 2. 0 3000. A proportion of these, about one-third, end at VS defects, from whence they return to AgZn defects 389 Luminescence and emit photons. The separation of the defect energy levels is approximately 3 eV, giving an emission at approximately 410 nm. The yellow emission is obtained by doping ZnS with CdS and employing the same silver activator. As CdS is 2. 0 added to the ZnS, the energy-level separation of the VS and AgZn defects decreases, and as the composition becomes richer in CdS the emission moves towards the red end of the spectrum, so that a continuum of þ þ þ colours can form between ZnS:Ag (blue), Zn0.68Cd0.32S:Ag (green), Zn0.5Cd0.5S:Ag (yellow) and þ þ Zn0.13Cd0.87S:Ag (red). Awidely used composition for yellow emission is Zn0.5Cd0.5S:Ag (Figure 9.19b). Colour TV uses similar phosphors for the blue and green emission, relying upon the same donor acceptor recombination, as above. For blue, silver-activated zinc sulfide (ZnS:Ag þ ) or zinc sulfide doped with AgCl to form ZnS:Ag þ ,Cl is used (Figure 9.19c and d). In this latter material, the important defects for colour 0 2 . emission are AgZn as before and chlorine ions (Cl ) substituted for sulfur (S ) to form ClS centres: ð Þ! 0 þ . AgCl ZnS AgZn ClS

The colour of the emission can be tuned somewhat by altering the defect energy, and for this the ZnS is . . sometimes doped with AlCl3, to introduce additional AlZn defects (Figure 9.19e). The transition from AlZn to 0 AgZn is at a slightly different wavelength and changes the hue of the display. þ 0 Thegreen phosphor is zinc sulﬁde doped with CuCl to introduce Cu ions onto Zn sites to form CuZn defects . and acceptor levels and ClS defects as before:

0 . CuCl ðZnSÞ!CuZn þ ClS

0 0 The CuZn defects give rise to higher energy levels than those introduced by AgZn , resulting in an emission centred at 530 nm, compared with approximately 410 nm for the blue emitter (Figure 9.19f). The colour can be tuned by incorporation of CdS, or other defects as described above. Early red phosphors for colour television also used ZnS-based materials, especially silver-doped zinc cadmium sulfide ((Zn,Cs)S:Ag þ ) described above. This colour was not satisfactory, and emission from Eu3 þ , similar to that used in fluorescent lamps (Section 9.4) was an early replacement. The first host matrix used was 3 3 þ YVO4. In this material, cathode ray energy was absorbed by the VO4 group and transferred to the Eu ions in 3 þ a charge-transfer process. This has been replaced by Y2O2S:Eu , which gives a brighter emission. Light emission is due to transitions between excited 5D levels and lower energy 7F levels that lie in the band gap of 5 ; 5 ; 5 ; 5 ; 5 the host. There are five of the former of importance, D0 D1 D2 D3 D4, and seven of the latter, 7 ; 7 ; 7 ; 7 ; 7 ; 7 ; 7 F0 F1 F2 F3 F4 F5 F6. To obtain the desired red colour output it is necessary to ensure that the 5 ! 7 dominant transition is D0 F2. (Although the terms suggest that this transition should give only one emission line, a small crystal-field splitting (Section 7.9) gives rise to a close doublet.) The result is a pair of lines at approximately at 612 and 628 nm (Figure 9.20). The limitation imposed upon the emission is achieved by a careful selection of the host material (Y2O2S is preferred for this reason) and by adjusting the concentration of 3 þ 5 Eu ions, which quench emissions from the D1 and higher D levels by cross-relaxation (Section 9.9). In working TV tubes the phosphor layers are backed by an aluminium film to increase the brightness of the image.

9.6.3 Other applications of cathodoluminescence

There are a number of other devices which use CRT technology widely. The most familiar are computer monitors, which are essentially television monitors, ﬂying spot scanners, oscilloscopes and radar screens. These need different decay characteristics than TV screens, and the phosphor technology in each is tailored to the exact requirements of the product. Colour and the Optical Properties of Materials 390

eV cmÐ1 O 2p - 4f band

5 40000

4 30000 5D 3 3 5 D2 Energy 20000 5D 5D 1 2 0

10000 1 7 F6 7 0 F0 Eu3+ (4f6)

þ Figure 9.20 Schematic energy-level diagram of Eu3 used as the red emitter in colour television CRTs. The 5 ! 7 important transition is D0 F2 with emission at approximately 612 and 628 nm

Cathodoluminescence has long been used in electron microscopes. In effect, a transmission electron microscope is rather like a long CRT. Electrons are emitted from an electron gun and traverse the specimen before hitting a ﬂuorescent screen coated with a cathodoluminescent phosphor which displays the image. The brightness of the image is a simple measure of the intensity of the electron beam. To maintain a good performance, the phosphor coating has to be replaced periodically, as the intense irradiation causes degradation of performance. Cathodoluminescence is also used as an analytical tool, particularly in scanning electron microscopes. In these latter instruments, an electron beam is scanned over the sample in a raster fashion. The sample can emit light by direct cathodoluminescence, or else cathodoluminescence can be generated from secondary electrons emitted under the primary beam. In both cases the cathodoluminescent spectra can be recorded and used for analytical purposes. In this way, a wide variety of opaque objects, ranging from archaeological or ﬁne art artefacts to semiconductor devices can be analysed using a nondestructive method.

9.7 Field-Emission Displays

Field-emission displays (FEDs; also called field-effect displays) use electron-excited phosphors as the light- emitting mode. They are similar to plasma displays, in that the display screen is composed of many tiny cells, but colour production in each cell is comparable to that in a CRT. In an FED, each cell contains a microscopic cathode. A voltage is applied to a cathode, which then emits electrons the process of field emission. The electrons are accelerated by the voltage differential and strike the phosphor as in a conventional CRT and, hence, produce light emission. The difficulty lies in the ejection of electrons from the cathode. The energy needed to force electrons out of a metal is called the work function. Electron emission in a normal CRT electron gun is usually accomplished by heating: the thermionic effect. This is not possible in FED. An alternative is to pull electrons out of the cathode using an electrostatic field. Under ordinary conditions, an extremely high electrostatic field is required. The problem is overcome in FEDs by making the cathode in the form of a sharp spherical tip. The static electric 391 Luminescence

ﬁeld F generated by an applied voltage V to a tip of radius r is:

V F ¼ r

Thus, the field is considerable if the tip has a radius of a few nanometres, even under the imposition of only a few volts. To capitalize on this fact, the cathode in each cell is made of a microscopically pointed spike, using a refractory metal such as molybdenum, carbon nanotubes or low work-function materials such as synthetic diamonds. Field emission then occurs even with a relatively low external voltage. The light emission is similar to that described in a CRT. Electrons, leaving the cathode, are accelerated by the electric field and strike a phosphor coated on a glass substrate with energy of about 100 eV.This principle is now actively explored for flat-panel displays, but none are yet in commercial production.

9.8 Phosphor Electroluminescent Displays

Electroluminescent displays containing a thin film of a phosphor, called thin-film electroluminescent (TFEL) displays, are, like FEDs, also akin to CRTs, in that the colour is generated by stimulation of a phosphor by energetic electrons. These devices find use as display panels, backlighting in products such as instrument panels, in flat-panel colour displays and in some aspects of lighting. The principle of operation involves the excitation of a chosen phosphor by high-energy electrons created within the phosphor film itself. Electrons enter the phosphor at the junction with a surface insulating coating and are accelerated under the influence of a high electric field. These electrons collide with the luminescent centres in the phosphor, transferring energy in the process. The excited luminescent centres then fall back to the ground state and release energy by light emission (Figure 9.21a). The most promising devices use AC supplies in a TFEL (ACTFEL) display. The device is built up in thin layers on a glass substrate (Figure 9.21b). A layer of a transparent electrical conductor, most often indium tin oxide, of about 400 nm thickness, is laid down first as one electrode. This is covered with about 400 nm of a transparent insulator. The active layer, about 700 nm of phosphor, and then another 400 nm layer of transparent insulator are then added. Finally, a 200 nm thick layer of aluminium is deposited on the stack. This serves as an electrode and also reflector. The display is viewed through the glass substrate, which acts as a protective surface. The accelerating field in these devices is of the order of 1 2 MV cm 1. This is generated by applying a lower voltage across a thin insulating layer, which acts as a capacitor. The whole arrangement is, in fact, a series of capacitors. This design is chosen because the high voltages which are needed in the phosphor layer are generated from low electrodevoltages by way of the capacitance of the thin insulating layers. To optimise the high fields in the phosphor, the dielectric needs to have a high relative permittivity (dielectric constant) and high breakdown strength, as well as being transparent to light. The ferroelectric barium titanate (BaTiO3) is often chosen. The most efficient electroluminescent thin-film phosphors consist of zinc sulfide containing manganese (ZnS:Mn2 þ ) as the luminescent centre. The Zn2 þ ions in the host are in tetrahedral coordination in both the cubic zinc blende form and the hexagonal wurtzite form. Mn2 þ ions readily replace Zn2 þ in the crystals. The 2 þ 5 6 6 ground state of the free Mn (d ) ion is S, which transforms to a single A1 energy level in the tetrahedral crystal field of the surrounding 4S2 ions in ZnS. The first free-ion higher energy term is 4G, which splits into 4 4 4 4 four in the tetrahedral crystal field, T1, T2, A1 and E (Figure 9.22; also see Section 7.9). The lowest of these 4 is T1 and the electroluminescent emission is due to the transition from this level to the ground state. The strong influence of the surrounding structure means that the spectrum, centred upon yellow, wavelength 585 nm, is broad. Optimal doping levels are close to 1 % Mn2 þ , as higher concentrations lead to concentration quenching of the emission. Colour and the Optical Properties of Materials 392

(a) insulator phosphor insulator

A e– photon

(b) light emission

glass substrate

indium tin oxide insulator phosphor

insulator aluminium Figure 9.21 TFEL displays. (a) Schematic representation of the process taking place in an electroluminescent material. Electrons enter the phosphor from an insulator–phosphor interface and accelerate under a high voltage. These transfer energy to luminescent centres A via collisions and these, in turn, lose energy by emitting light. (b) Idealised electroluminescent thin-ﬁlm unit

A number of other colours can be generated using doped ZnS containing Cu þ ,Al3 þ and Cl as described above. ZnS:Cu þ ,Cl gives out blue light with a wavelength near to 460 nm or green light with a wavelength near to 510 nm, depending upon the concentration of Cl present. The defects formed are identical to those described earlier. The presence of Al3 þ donors is used to adjust the exact emission colour. Colours can also be generated by the incorporation of lanthanoids into the host phosphor. For example, red emission is produced by calcium sulﬁde doped with europium (CaS:Eu2 þ ). The Eu2 þ ions have a 4f7 electron 8 conﬁguration with a ground state S7=2, and an upper energy band corresponds to the transfer of an electron into

4 T2 4 E1 4 G 4 A1 4 T1

~585 nm

6 6 ground state S A1 free ion ion in ZnS þ Figure 9.22 The energy levels of Mn2 free ions and in the tetrahedral crystal ﬁeld of ZnS (schematic) 393 Luminescence

4f7 - 5d band (a) (b) (c)

5D 1 3 D2 1 4f6 - 5d band G4 5 790 nm D4 3F 3 2 623 nm F3 3 H4 ~640 nm 589 nm 650 nm 3 489 nm H5 450 nm 7 3 F0 F4

8 7 3 S7/2 F6 H6 Eu2+ (4f7) Tb3+ (4f8) Tm3+ (4f12)

(d) 5d band

~445 nm ~459nm

2 F7/2 2 F5/2 Ce3+ (4f1)

þ þ þ þ Figure 9.23 The energy levels of (a) Eu2 , (b) Tb3 , (c) Tm3 and (d) Ce3 in TFEL host structures (schematic) the outer 5d orbital to give a configuration 4f6 5d. The colour-generating transition is from this upper band back to the ground state (Figure 9.23a). At first sight this is surprising, as the usual blue tricolour lamp phosphor, 2 þ BaMgAl10O17:Eu , has a maximum at 450 nm (Section 9.4). However, the exact position of the upper energy band depends upon the interaction of the d orbitals with the surrounding crystal, and in ZnS the softer bonding gives a broad emission centred close to 640 nm. A strong green line emission is produced by zinc sulfide doped with terbium (ZnS:Tb2 þ ). The green 5 7 emission is at wavelength close to 545 nm, mainly from a D4-- F5 transition (Figure 9.23b). Three other peaks 5 7 5 7 5 7 of lesser intensity occur: D4-- F6, 489 nm; D4-- F4, 589 nm; D4-- F3, 623 nm. Note that these emission lines are very similar to those given out from oxide host crystals, because the 4f energy levels are well shielded and the f f transitions are thus insensitive to the surroundings. Blue emission still poses a problem for these displays, but one material that has been used is zinc sulfide doped with both thulium and fluorine (ZnS:Tm3 þ ,F ). In this latter case the thulium ions (Tm3 þ ) occupy positions normally filled by zinc (Zn2 þ ) ions. The fluoride (F ) ions are needed to compensate for the excess charge on the thulium ions, (Tm3 þ þ F ) being equivalent to (Zn2 þ ). The emission, at 450 nm, is sharp, due to an f f transition 1 ! 3 G4 H6 (ground state). However, the emission is also weak and much energy is given out in the far red due to 1 ! 3 1 ! 3 G4 F4 and in the infrared due to G4 H5 transitions (Figure 9.23c). Superior blue emission has been 1 3 þ obtained from the thiogallates CaSr2S4,SrGa2S4 and BaGa2S4 doped with the 4f ion Ce . The transitions Colour and the Optical Properties of Materials 394

100

40 Relative intensity 20

400 500 600 700 Wavelength / nm þ þ Figure 9.24 The emission spectrum of the stacked TFEL phosphors ZnS:Mn2 and SrS:Ce3 , which give white output to the eye

1 2 2 between the excited 5d state and the 4f ground-state doublet F5=2 and F7=2 are both in the blue region of the spectrum (Figure 9.23d). Because of the involvement of the more exposed 5d electron energy levels, the emission is quite broad, centred at 459 nm for the Ca compound and 445 nm for the Sr and Ba phases. White-light emission can also be achievedusing combinations of luminescent centres. Oneway in which this has been achieved is to use stacked layers of ZnS:Mn2 þ and SrS:Ce3 þ . These broad-emission band phosphors combine to produce a light with peaks in the green and yellow that appear white to the eye (Figure 9.24). Full-colour displays can be constructed in a number of ways. The simplest, conceptually, is to use a white- emitting phosphor such as that just described and to incorporate a coloured ﬁlter into the device (Figure 9.25a). An alternative is to build up subarrays of pixels with red, blue and green phosphor subpixel units (Figure 9.25b). By varying the voltage distribution between the aluminium and indium tin oxide electrodes, any of the red, blue or green phosphors can be excited to luminesce. Other device geometries have been used, including stacks of single emitting devices.

9.9 Up-Conversion

The detection of infrared radiation and subsequent conversion to visible has many possible applications, encompassing the generation of white light from LEDs (Chapter 10) and the study of nocturnal mammals. One method of achieving this objective is variously known as frequency up-conversion, anti-Stokes fluorescence or cooperative luminescence. In this effect, low-energy radiation, typically in the infrared, is ‘up-converted’ into visible radiation. Up-conversion is thus the opposite of photoluminescence, in which high-energy ultraviolet radiation is ‘down-converted’ into visible light. Note that up-conversion is distinct from frequency doubling, which uses nonlinear polarisation rather than luminescence (Section 4.9). The up-conversion efficiency of a system can be defined as the ratio:

power emitted ðvisibleÞ Efficiency ¼ power absorbed ðinfraredÞ

In general, the up-conversion efficiency is low and varies with the concentration of the activator and sensitiser ions. In general, low concentrations of the active ion, of the order of 1 %, are used. At these concentrations, the 395 Luminescence ions form point defects well isolated from each other. At higher concentrations, dopant ions tend to cluster and other energy loss mechanisms interfere with up-conversion. The exitance of the up-converted output Iup is related to the irradiance of the exciting radiation Iex by the formula: / n Iup Iex where n is the number of photons absorbed per up-converted photon emitted. A graph of log Iup versus log Iex will give information on the mechanism of the process, as detailed below. (The data given in the literature often uses ‘intensity’ in arbitrary units for the quantities Iup and Iex, or ‘intensity’ in arbitrary units for Iup and ‘pump power’ for Iex.) The majority of studies of up-conversion have dealt with the behaviour of lanthanoid ions, especially Er3 þ , Tm3 þ ,Ho3 þ and Yb3 þ . A number of host structures have been used for these ions, including binary oxides Y2O3,Gd2O3 and ZrO2, perovskite structure BaTiO3, SrTiO3 and PbTiO3, fluorides such as NaYF4 and oxide and oxyfluoride glasses.

9.9.1 Ground-state absorption and excited-state absorption

The energy for up-conversion can be gained by the active ion in several ways, and there are often many competing energy transfer processes taking place during up-conversion (Table 9.2). In principle, the simplest

(a) light emission

indium tin oxide electrodes

insulator

white phosphor

insulator

aluminium electrodes glass substrate

Figure 9.25 ACTFEL device structures (schematic): (a) colour display element using a white-emitting phosphor and colour ﬁlters; (b) colour display element using phosphor subpixels emitting red, blue and green Colour and the Optical Properties of Materials 396

(b) aluminium electrodes

insulator

red, blue, green phosphors

insulator

indium tin oxide electrodes

glass substrate

light emission

Figure 9.25 (Continued ) is for the activator ion to pick up photons in two distinct steps. The ﬁrst photon excites the ion from the ground state to an excited energy level, a process referred to as ground-state absorption (GSA). A subsequent photon is then absorbed to further promote the excited ion to a higher energy level again excited-state absorption (ESA). 3 þ 3 þ The oxide CeO2 doped with approximately 1 % Er exhibits up-conversion in this way. The Er ions substitute for Ce4 þ to form a low concentration of Er3 þ ions randomly distributed within the oxide matrix. Irradiation with near-infrared photons with a wavelength close to 785 nm, the pump wavelength, excites the 3 þ 4 4 Er ions from the I15=2 ground state to the I9=2 level, that is; a GSA mechanism:

4 4 I15=2 þ hn ð785 nmÞ! I9=2

4 4 These ions lose energy nonradiatively to phonons (lattice vibrations) to reach the I11=2 and I13=2 energy levels (Figure 9.26a):

4 4 4 I9=2 ðErÞ! I11=2 ðErÞþ I13=2 ðErÞþphonons

Ions in both these levels are further excited by an ESA mechanism via the same pump wavelength to gain 4 energy from the irradiating 785 nm radiation. For example, those in the I11=2 energy level are excited to the 4 F3=2; 5=2 doublet:

4 4 I11=2 þ hn ð785 nmÞ! F3=2;5=2 397 Luminescence

2 4 4 These states subsequently relax via internal energy loss to the H11=2; S3=2 and F9=2 energy levels (Figure 9.26b):

4 2 4 4 F3=2;5=2 ! H11=2 þ S3=2 þ F9=2 þ phonons

4 2 The ions in the I13=2 energy level follow a similar path, being excited to the H11=2 energy level:

4 2 I13=2 þ hn ð785 nmÞ! H11=2

4 4 then subsequently relax to the S3=2 and F9=2 energy levels (Figure 9.26c):

2 4 4 H11=2 ! S3=2 þ F9=2 þ phonons

(a) 25 2 H9/2 4 F3/2;5/2 –1 4F 20 7/2 2 4 H11/2 S3/2 4 15 F9/2 4 I9/2 4 10 I11/2 Energy / 1000 cm

4 I13/2 5 785 nm hν GSA 4 0 I15/2 Er3+ GSA + relaxation

(b) 2 H9/2 4 F3/2;5/2 4F 2 7/2 H11/2 4 S3/2 785 nm 4 F9/2 4 I9/2 ν 4 h ESA I11/2

4 I13/2

4 I15/2 ESA + relaxation

3 þ Figure 9.26 Up-conversion in CeO2 doped with Er (CeO2:Er): (a) GSA plus decay; (b) ESA plus decay; (c) ESA plus decay; (d) emission; (e) emission spectrum following up-conversion Colour and the Optical Properties of Materials 398

4 4 F9/2 F9/2

4 785 nm 4 I9/2 I9/2 4 4 I11/2 I11/2 ν 4 h ESA 4 I13/2 I13/2 526 nm (green) 547 nm (green) 658 nm (red)

4 4 I15/2 I15/2 ESA+relaxation emission

(e) I → 3/2 15/2 S 44 I 4 I Intensity (arbitrary units) →

→ 9/2 15/2 F 4 11/2 15/2 H 24 500 600 700 Wavelength / nm

Figure 9.26 (Continued )

2 4 4 The result of this is that the energylevels H11=2; S3=2 and F9=2 are populated tovarying degrees, depending upon the precise details of the excitation and relaxation steps. Subsequent loss of energy from these levels gives rise to green and red emission (Figure 9.26d):

2 4 H11=2 ! I15=2 þ hn ð526 nm; greenÞ 4 4 S3=2 ! I15=2 þ hn ð547 nm; greenÞ 4 4 F9=2 ! I15=2 þ hn ð658 nm; redÞ

The up-conversion spectrum consists of three major peaks (Figure 9.26e). þ All up-conversion spectra from Er3 (including those using different mechanisms, below), are similar, but the relative intensities and positions of the three peaks vary with concentration of activator ions, sensitiser ions and the nature of the host matrix. Some of these aspects are outlined in the following sections. 399 Luminescence

9.9.2 Energy transfer

When the concentration of the dopant Er3 þ rises, other processes become important. In this section the two simplest of these are described. Pump energy can be picked up by a sensitiser and transferred directly to the emitter; a process called energy transfer (ET). (This is also what happens in a phosphor containing a sensitiser.) Energy transfer can be illustrated by up-conversion in a host containing the co-dopants Yb3 þ /Er3 þ , which give a strong green emission. The ion that absorbs the incoming infrared radiation is the Yb3 þ ion, which then þ þ transfers energy to the Er3 active ion. Generally, the concentration of the absorbing Yb3 centres is about 20 %, while the concentration of the activator Er3 þ ions is about 1 %. The energy levels of the infrared radiation suited to the energy transfer process match the Yb3 þ ion energy 2 2 transition from the ground state F7=2 level to the F5=2 level and lasers with an output of 975 nm are usually 4 4 3 þ employed. This pump energy also matches the I15=2 to I11=2 GSA transition of Er centres, but energy transfer from the Yb3 þ centres dominates the process:

2 2 GSAðYbÞ F7=2 ðYbÞþhn ð975 nmÞ! F5=2 ðYbÞ 2 4 2 4 ET F5=2 ðYbÞþ I15=2 ðErÞ! F7=2 ðYbÞþ I11=2ðErÞ

4 This is followed by nonradiative relaxation of some ions to the I13=2 level (Figure 9.27a):

4 4 I11=2 ðErÞ! I13=2 ðErÞþphonons

The second stage in the excitation process can use any of three mechanisms. Two are similar to those just described; that is, further gain of energy from Yb3 þ (ET) or absorption of a photon (ESA):

2 4 2 4 ET F5=2 ðYbÞþ I11=2 ðErÞ! F7=2 ðYbÞþ F7=2 ðErÞ 4 4 ESA I11=2 ðErÞþhn ð975 nmÞ! F7=2 ðErÞ

2 4 Some ions relax to the H11=2 (Er) and S3=2 (Er) levels (Figure 9.27b and c):

4 2 4 4 F7=2 ðErÞ! H11=2 ðErÞþ S3=2 ðErÞþ F9=2 ðErÞþphonons

The third mechanism that operates involves energy transfer between two excited Er3 þ ions in a process called cross-relaxation (CR), which results in further excitation of one ion and loss of energy of the other (Figure 9.27d):

4 4 4 4 CR I11=2 ðErÞþ I11=2 ðErÞ! F7=2 ðErÞþ I15=2 ðErÞ

4 2 The populated F7=2 (Er) level is able to lose energy nonradiatively as above. The end result is that the H11=2 4 4 (Er), S3=2 (Er) and F9=2 (Er) energy levels are populated. 4 4 The F9=2 (Er) level can also be populated by transitions from the I13=2 (Er) level (which itself was populated 4 by nonradiative relaxation from the I11=2 (Er) level) in these three ways (Figure 9.27e):

4 4 ESA I13=2 ðErÞþhn ð975 nmÞ! F9=2 ðErÞ 2 4 2 4 ET F5=2 ðYbÞþ I13=2 ðErÞ! F7=2 ðYbÞþ F9=2 ðErÞ 4 4 4 4 CR I13=2 ðErÞþ I11=2 ðErÞ! F9=2 ðErÞþ I15=2 ðErÞ Colour and the Optical Properties of Materials 400

2 4 4 The end result is that the energy levels H11=2; S3=2 and F9=2 are populated to varying degrees, giving rise to the same photon emissions as detailed above (Figure 9.26e):

2 4 H11=2 ðErÞ! I15=2 ðErÞþhn ð525 nm; greenÞ

4 4 S3=2 ðErÞ! I15=2 ðErÞþhn ð550 nm; greenÞ

4 4 F9=2 ðErÞ! I15=2 ðErÞþhn ð655 nm; redÞ

A maximum efficiency is observed with concentrations of about 1 3 % of the active centre. Note that these are only an outline of the many processes that can occur. It is easy to imagine back transfer þ þ þ þ from Er3 to Yb3 and Yb3 Yb3 energy transfer, both of which will lower the efficiency of the process. In addition, increasing interactions between both lanthanoid ions can lead to cluster formation. In effect, this changes the site symmetry and surrounding matrix experienced by the ions, again limiting the efficiency.

(a)

) 25 H 9/2 –1 4 F 3/2;5/2 4 20 F 7/2 4 2 S 3/2 H11/2 4 15 F 9/2 4 I 9/2 Energy / 1000 cm 2 4 10 F5/2 I 11/2

ET 4 I13/2 5 975 nm hν GSA 2 4 0 F7/2 I15/2 Yb3+ Er 3+

2 H9/2 (b) 4 F3/2;5/2 4 F 7/2 2 4 H11/2 S3/2 4 F9/2 ET 4 I9/2 2 4 F5/2 I11/2

4 I 975 nm 13/2

2 4 F7/2 I15/2 Yb3+ Er3+ þ þ þ þ Figure 9.27 Up-conversion in a matrix containing the co-dopants Yb3 /Er3 : (a), (b) ET from Yb3 to Er3 ; þ þ (c) ESA in Er3 ; (d) CR in Er3 ; (e) ET, CR and ESA 401 Luminescence

4 4 F9/2 F9/2

hν ESA 4 4 I9/2 I9/2 4 CR 4 I11/2 I11/2

4 4 I13/2 I13/2

4 4 I15/2 I15/2 Er 3+ Er3+

(e) 2 H9/2 4 F3/2;5/2 4F 7/2 2 4 H11/2 S3/2

4 F9/2

4 I9/2 2 F 4I 5/2 hν ESA 11/2 ET CR Energy 4I 975 nm 13/2

2 4 F7/2 I15/2 Yb3+ Er 3+

Figure 9.27 (Continued )

9.9.3 Other up-conversion processes

Other up-conversion processes are known. The blue emission from a Yb3 þ /Tm3 þ couple in which the active emitters are Tm3 þ centres is mainly due to the efﬁcient triple excitation ET process from Yb3 þ centres (Figure 9.28), although CR and other complexities cannot be ignored in a detailed interpretation of the process. The interest in this process is partly because, if combined with a Yb3 þ /Er3 þ couple in the same host lattice, the red, green and blue emissions produce a white light output. Two-frequency up-conversion has been investigated using Pr3 þ defects in a ﬂuoride glass matrix. 1 Illumination with one pump wavelength, 1014 nm, results in GSA to the metastable G4 energy level. No further excitation is possible with this pump, but simultaneous irradiation with a second appropriate pump 3 wavelength, 850 nm, further excites the GSA centres via ESA to the P3 level. The doubly excited ions lose 3 3 energy by nonradiative decay to the P0 level. These then drop to the H6 level and emit red light (Figure 9.29). Up-conversion and visible output only takes place at the intersection of the two beams. Note that these are only an outline of the many processes that can occur in a phosphor. In systems that rely upon a sensitiser, energy transfer must take place between the two centres. Energy transfer in the reverse Colour and the Optical Properties of Materials 402

cm–1 eV

30000 3.5

25000 1D 3.0 2

1 G4 20000 2.5 Energy

3 F2 2.0 3 15000 F3

3H 1.5 4 ET 10000 3 1.0 H5 2 ET F5/2

3 5000 F4 0.5 ET 975 nm

2 3 F7/2 H6 Yb3+

Tm3+ (4f 12) þ Figure 9.28 Schematic energy-level diagram for Tm3 . Red arrows indicate excitation, via energy transfer þ from Yb3 sensitiser ions. Dotted arrows show nonradiative losses and the blue arrow indicates the main blue output direction may happen as the concentrations change, which leads to concentration quenching (Section 9.3). Back energy transfer from Er3 þ to Yb3 þ and Yb3 þ to Yb3 þ energy transfer both lower the efﬁciency of the process, as does the presence of defects in the phosphor matrix. In addition, the degree of phosphor crystallinity and particle size are important. For this reason, many compositions and dopant levels are explored systematically before an optimum composition and preparation route are achieved.

9.10 Quantum Cutting

In up-conversion, several low-energy photons are processed within a luminescent matrix to give out one higher energy photon, typically infrared to visible. Quantum cutting is the reverse of this, as one high-energy photon is processed (i.e. cut) to give out several lower energy photons, typically ultraviolet to visible. One aim of this work is to improve the efﬁciency of phosphors in ﬂuorescent lamps. Here, the driving force is to eliminate the mercury vapour component of the lamp and replace it with less toxic gases, such as xenon. However, the main Xe emissions are at 147 and 172 nm, compared with 254 nm of mercury. Thus, new phosphors need to be found that are stable under these intense ultraviolet rays and can compete with mercury vapour lamps in terms 403 Luminescence

cm–1 eV

30000 3.5

25000 3.0

3 P3

20000 2.5 3 P0

1D Energy 2.0 2 15000 850 nm pump red emission 1.5

10000 1 G4 1.0

3 1014 nm F3 5000 pump 3 3 F2 0.5 H6 3 H5

3 H4 Pr3+ 4f3 þ Figure 9.29 Schematic two-frequency up-conversion process in Pr3 resulting in the production of red light of luminosity. Quantum cutting is valuable in this context, as one input ultraviolet photon can be cut to yield several visible photons. Further applications in colour plasma display panels are also being explored for this technique. þ There are two main mechanisms for quantum cutting. The ﬁrst is photon cascade emission, typiﬁed by Pr3 (4f3) ions. Initial absorption of high-energy 185 nm ultraviolet photons causes excitation to the 4f2 5d band 3 þ 1 of Pr (Figure 9.30). Subsequent relaxation takes the ion to the S0 level. Thereafter, the transitions giving rise to visible output are:

1 ! 3 ; 3 ! 3 S0 P3 at 400 nm then P0 H4 ground state at 480 nm 1 ! 1 ; 1 ! 3 S0 D2 at 330 nm then D2 H4 ground state at 605 nm

3 þ The second mechanism explored involves cross-relaxation involving Tb ions, in, for example, GdPO4: Tb3 þ .AswithPr3 þ , initial absorption of the ultraviolet photons causes excitation of the active Tb3 þ (4f8) ions to the 4f7 5d band (Figure 9.31). Absorption of 210 nm ultraviolet photons leads to the following steps: 3 þð Þð 7 Þ!5 3 þð Þ 5 7 1. Tb 1 4f 5d D3 and via cross-relaxation simultaneously gives Tb 2 D4 F6 ground state, 5 ! 7 5 ! 7 then D4 F5 plus green emission at approximately 550 nm (Figure 9.31a). Transitions D4 FJ, with J ¼ 3, 4 and 5, also occur with lesser intensity. Colour and the Optical Properties of Materials 404

3 þð Þ 5 ! 5 ! 7 2. Tb 1 D3 D4 nonradiative transition F5 plus green emission at approximately 550 nm 5 ! 7 ¼ (Figure 9.31b). Transitions D4 FJ, with J 3, 4 and 5, also occur with lesser intensity. Other energy transfer processes also occur involving the host Gd3 þ ions, but these do not give rise to strong emissions in the visible.

4f2 - 5d band cmÐ1 eV

6.5

50000 6.0 1 S0 45000 5.5

400 nm 40000 5.0

4.5 35000 330 nm 185 nm 4.0

30000 3.5 Energy

25000 1 3.0 I6 3 P3

20000 2.5 3 P0

1D 2.0 2 15000 605 nm 1.5

10000 1 G4 1.0 480 nm 3 F3 5000 3 3 F2 0.5 H6 3 H5

3 H4 Pr3+ 4f3

Figure 9.30 Quantum cutting of one 185 nm ultraviolet photon to give two photons at 400 and 408 nm or at 330 and 605 nm. Both processes occur 405 Luminescence

4f7 - 5d band (a) eV cm–1

5 40000

4 30000 5 CR D3 3 5 Energy 20000 D4 2 550 nm 10000 1 7 F0

7 0 F6 Tb3+ (1) (4f 8 ) Tb3+ (2)

7 (b) 4f - 5d band

5 D3

5 D4

550 nm

7 F0

7 F6 Tb3+ (1) (4f 8 ) þ Figure 9.31 Quantum cutting of one 210 nm ultraviolet photon into two 550 nm green photons by Tb3 ions (schematic)

9.11 Fluorescent Molecules

9.11.1 Molecular ﬂuorescence

The energy levels giving rise to fluorescence in molecules are most often of associated with the HOMO LUMO pair of molecular orbitals that are formed by delocalised electrons. This implies that many of the aromatic compounds, conjugated molecules and dyes described in Chapter 8 are also fluorescent, and many fluorescent molecules are referred to as fluorescent dyes. Typical examples include the anthraquinones, xanthenes, cyanins, phthalocyanines and porphyrins. Fluorescein (Figure 9.3)is typical of thesegroups, with an absorption Colour and the Optical Properties of Materials 406 peak at 495 nm and an emission peak at 519 nm, giving rise to the characteristic yellow green hue of materials coloured with this substance. Another group of fluorescent molecules that is being actively explored consists of an electron-accepting group (a Lewis acid, such as phenol) connected to an electron-donating group (a Lewis base, such as methylamine) by a series of aromatic (benzene) rings. These types of compound are known generally as donor p-bridge acceptor molecules. They display intense fluorescence; and because of the exposed donor and acceptor groups, they often show pronounced solvatochromism. There is also considerable work in progress on the incorporation of luminescent molecules into polymers, thin films and liquid crystals, for potential optoelectronic applications. In addition, fluorescent tags attached to molecules can be exploited to follow the course of chemical reactions, including catalysis, in which the amount of catalyst is small and the sensitivity of the fluorescence technique is vital. The schematic energy-level diagram of a typical molecular fluorophore (Figure 9.32a) shows that the absorption of energy is from the lowest vibrational level (J ¼ 0) of the ground state to the various vibrational energy levels of the LUMO, whilst the emission is from the lowest vibrational level (J ¼ 0) of the LUMO to the various vibrational levels of the ground state. The absorption transition from J ¼ 0 ground state to J ¼ 0 excited state is at the same energy as the emission from J ¼ 0 excited state to J ¼ 0 ground state. From this it follows that, ideally, the emission and absorption curves from a molecule are approximately mirror images around this energy (or wavelength). In ideal cases, the fine structure peaks on the absorption spectrum indicate the vibrational energy-level spacing of the excited state and the fine structure peaks on the fluorescence spectrum indicate the vibrational energy-level spacing of the ground state (Figure 9.32b).

(a) excited state 1 A*

J=0

absorption fluorescence

J=0 ground state 1 A

(b) Stokes shift

J = 0 to J = 0 transition excited state vibrational levels ground state vibrational levels

absorption fluorescence

Wavelength

Figure 9.32 Idealised absorption and ﬂuorescence from a molecule: (a) energy level (Jablonski) diagram; (b) absorption and emission spectra 407 Luminescence

9.11.2 Fluorescent proteins

Although many fluorescent molecules are important, there is some justification to the argument that fluorescent proteins are the most important, particularly in the light of current biological research. Indeed, ‘life may be defined as the ordered interaction of proteins’ (Further reading, Section 9.17, D. Whitford, p. 2). Moreover, the ‘central dogma’ of molecular genetics, viz. DNA makes RNA makes protein, puts the importance of fluorescent proteins into context. Using standard laboratory techniques, fluorescent proteins can be incorporated into cellular pathways via DNA modification. The course of action of the subsequent fluorescent engineered molecules can then be observed using fluorescence microscopy to study gene expression, protein protein interactions and cell reaction pathways in a multiplicity of organisms, from the simplest to the most complex. The first fluorescent protein to be discovered, green fluorescent protein (GFP), was isolated from a coelenterate medusa (jellyfish) Aequorea victoria. It has a barrel-like structure composed of 11 antiparallel b-sheets approximately 3 nm in diameter and 4 nm in length (Figure 9.33a). The fluorophore is part of a single a-helix positioned in the centre of the barrel. It is formed from three adjacent amino acids in the helix, serine (Ser 65), tyrosine (Tyr 66) and glycine (Gly 67), where the numbers refer to the position of the amino acid in the chain. This triplet of amino acids occurs commonly in proteins; the important difference between the sequence in GFP and other proteins is in the location of the group. This is such as to allow the amino acids to link up in a specific cyclic way to form the fluorophore with a notable sequence of conjugated bonds (Figure 9.33b and c). Naturally occurring or ‘wild-type’ GFP (wtGFP) absorbs mainly at 395 nm (via a protonated form of the fluorophore), and to a lesser extent at 475 nm (via a deprotonated form of the fluorophore). These two forms are present in varying amounts depending upon the pH and temperature of the surroundings. The emission is at 509 nm, irrespective of the absorption, and the quantum efficiency is about 0.75. Proteins are folded and coiled in very specific ways, and a mistake in this will prevent the protein from carrying out its normal cellular function. This specific folding is usually achieved with the help of other cellular molecules produced in the cell. Remarkably, the complicated folding required to produce functioning GFP occurs without the necessity of co-reactants only found in the cells of the living animal, so that the fluorescent protein can be made in the laboratory and fused to a variety of enzyme and other protein targets so as to monitor cell processes using fluorescence microscopy. Despite this enormous advantage, there are drawbacks towtGFP. First, the complex folding needed occurs efficiently at 28 C, the normal ambient temperature encountered by A. victoria, but not very well at 37 C, the typical mammalian cell temperature. Second, the two absorption peaks are inconvenient. To offset these disadvantages, a variety of mutations have been made to the wtGFP. These changes are often brought about by just one or two modifications in the amino acid sequence making up the protein. At present, fluorescent proteins that form at 37 C and fluoresce with a variety of colours from blue (BFP), cyan (CFP), green (GFP) and yellow (YFP) are available. However, the GFP basic structure cannot be modified to yield a red fluorescing form. This gap has been filled by the isolation of a red fluorescent protein from corals the protein that gives many corals a pink tone. This material, initially isolated from the coral Discosoma striata,is known as DsRed, and has an emission peak at 583 nm. As with the GFP family, the DsRed family have also been chemically modified, so that a considerable number of red and orange fluorescent proteins are now commercially available. Despite the enormous usefulness of these families of fluorescent proteins, they suffer from a significant drawback. They need to be excited with radiation in the ultraviolet blue green region of the spectrum. These wavelengths do not penetrate tissues, and so most studies using them are confined to thin reaction volumes in vitro (literally, in glass, i.e. on glass microscope slides or shallow dishes). A new group of fluorescent proteins has now been produced that absorb infrared radiation and emit in the near infrared or deep red. The initial fluorescent protein was obtained from the bacterium Deinococcus radiodurans, which, as its name suggests, is able to survive in extreme environments. The great advantage of these fluorescent molecules is that the Colour and the Optical Properties of Materials 408 absorption and emission wavelengths readily penetrate mammalian tissue and bone. This means that studies can be carried out on living organisms buried deep inside the body. For example, malignant tumours can be tagged with fluorescent proteins and the processes taking place at a cellular level can be imaged using infrared detectors. The increasing importance of fluorescent proteins has become apparent in the last few years, and progress on the use of these molecules will undoubtedly continue unabated.

(b) O

tyrosine (Tyr) HO CH2 CH C OH

NH2 O

serine (Ser) HO CH2 CH C OH

NH2

glycine (Gly) CH2 C OH

NH2

Figure 9.33 GFP: (a) barrel-like structure of the protein; (b) molecular building blocks of the ﬂuorophore of GFP; (c) the structure of the ﬂuorophore of GFP 409 Luminescence

tyrosine (Tyr 66)

CH2

C O

N C C N O

HN CH C C

H2 glycine (Gly 67) CH

serine (Ser 65) 2 OH

Figure 9.33 (Continued )

9.11.3 Fluorescence microscopy

A principal application of molecular fluorescence is in fluorescence microscopy. The first fluorescence microscope was invented in 1911 and the first practical epi-fluorescence microscope in 1929, and thereafter fluorophores (also referred to as fluorochromes or fluorescent dyes) were used to stain tissues andbacteria before observation. Since then the technique has become a major instrument in the life sciences. It is powerful because, in a normal optical microscope, incident light is scattered by the object and then collected and passed to the observer via the optical train of the instrument. However, the fluorophore emits light, rather than just scattering incident light, and for this reason it is possible to investigate submicroscopic cellular components. Moreover, the realisation that a combination of fluorescent dyes and sophisticated imaging techniques would make it possible to surpass the conventional diffraction limit of an optical microscope has now led to the production of images of subcellular structures with detail resolved far below that thought possible even a few years ago. At its most basic, there are several distinct steps to obtaining fluorescence microscope images. First, the component of interest must be linked to a fluorophore if it is not self-fluorescent. There are a large number of commercial products available for this, comprising (i) small molecules such as fluorescein derivatives, made up of 30 or so atoms, (ii) fluorescent proteins, which are large molecules, made up of thousands of atoms, and latterly (iii) quantum dots, which are 10 100 atoms in size. Many of these are highly specific, and this variety makes it possible (in theory) to study a wide range of cellular processes simultaneously. Second, these markers must be inserted into the host tissue. This may involve the temporary modification of the fluorescent molecule to enable it to penetrate living tissue, after which the change is reversed within the cell so that the fluorescent form is regenerated. Of course, it is important that the fluorophore is neutral with respect to the biology that is being investigated. Some nanoparticles are toxic, for instance, which rules out use in live cells unless they are treated to avoid this difficulty. Note that the fluorescence observed may also reflect the nature of the cell fluids. For example, if a solvatochromic fluorophore is exposed to a watery cell fluid it may have a different fluorescent wavelength than if is in encapsulated in the interior of a hydrophobic region. However, these problems are often bypassed, and in such cases several different molecules can be used simultaneously to observe separate cellular functions simultaneously by recording the different fluorescent wavelengths emitted. Colour and the Optical Properties of Materials 410

Third, the treated material must be illuminated with a powerful source of exciting radiation, usually in the ultraviolet or blue region of the spectrum. The relatively weak fluorescence signal must be separated from the powerful beam used to excite the molecules and then be observed and recorded. This task is made more difficult by the fact that the contrast of the fluorescence is diminished by both Rayleigh and Mei scattering, which occurs from many of the organelles in living matter and can seriously degrade the image. (The technicalities of operation of a fluorescence microscope can be best appreciated by reference to manufacturer’s literature; see this chapter’s Further Reading). The likelihood of absorption of a photon of the exciting radiation by a fluorophore is quantified by the attenuation coefficient (formerly extinction coefficient) of the dye. Clearly, a high attenuation coefficient is a necessity. Similarly, the fluorescence quantum yield must be as close to unity as possible, to ensure that a light signal can be detected. One advantage of fluorescence microscopy is that it offers the opportunity of imaging single molecules. It is achieved because of the advent of strongly fluorescent molecules, particularly of the donor p-bridge acceptor type, that are able to emit 106 or more photons over a period of minutes. A remarkable example of this ability is demonstrated in experiments to determine how DNA molecules respond to stretching. This is of considerable relevance, as cellular life processes are centred upon the coiling, folding and unzipping of double-stranded DNA (see this chapter’s Further Reading). Forster€ resonant energy transfer (FRET) (Section 9.3), first explored in biological microscopy in the mid 1970s, is also a widely used technique. In this application, both the absorbing molecule A and the receiving molecule Q are designed to fluoresce efficiently at different wavelengths. This means that, in the absence of any energy transfer, fluorescence will be characteristic of A. If, however, A and Q approach close enough for energy transfer to occur, the fluorescence from Awill be partly or completely quenched, while fluorescence from Q will appear. In this way, cellular processes such as protein folding can be observed. A protein containing two fluorescent centres, A and Q, will show fluorescence from A if the folding does not bring the centres into coincidence, while fluorescence from Q will appear if the centres are juxtaposed after folding. Similarly, processes taking place across cell walls can be investigated. If a fluorescent molecule A is bound to the external surface of a cell wall and a fluorescent molecule Q is introduced or formed within the cell, fluorescence from Q will be seen as a result of FRET if Q is attached or very close to the internal cell wall. As can be imagined, there are many variations on this technique that are now in use. Another property of fluorescent molecules that is used in the study of cellular dynamics, including molecular diffusion, is photobleaching. Under the intense irradiation needed to image fluorescent molecules successfully in a microscope, many of the molecules decompose. The irradiated area (volume) then ceases to fluoresce and photobleaching is said to have occurred. In a dynamic situation, as in diffusion, for example, new fluorescent molecules can penetrate into the bleached region. A measurement of the rate of this recovery will give information about the mechanism of the molecular mobility taking place.

9.11.4 Multiphoton excitation microscopy

Multiphoton excitation microscopy, which in practice relies mainly upon two-photon excitation, is a complementary technique to fluorescence microscopy. Although the theory of two-photon absorption was worked out in 1931, the application to microscopy had to wait until the 1990s and the availability of lasers that could deliver the required light intensities. The recorded signal is again fluorescence, involving a fluorophore in the sample. However, in multiphoton excitation microscopy the fluorophore is excited by the simultaneous multiple absorption of low-energy photons. In this respect it resembles up-conversion, but is quite distinct and should not be confused. The fluorophore is able to pick up two or more photons to bridge the energy gap between the ground state and the fluorescent excited state without the necessity of populating or involving intermediate energy levels (Figure 9.34). For example, the simultaneous absorption of two infrared photons of 411 Luminescence

(a) excited state A* (b) excited state A*

ν E = h 1 ν E = 2 h 1 ν ν E = h 2 E = h 1 ν E = h 1

ground state A ground state A fluorescence up-conversion

ν E = h 1

ν E = 2h 1 ν E = h 1

ground state A two-photon fluorescence Figure 9.34 A comparison of the absorption mechanisms for (a) normal ﬂuorescence, (b) up-conversion and (c) two-photon ﬂuorescence

1050 nm can be used to excite a 525 nm absorbing fluorophore. A considerable advantage of the technique is that it allows access to ultraviolet absorbing fluorophores. In practice, it is difficult to build satisfactory optical systems for use at many ultraviolet wavelengths. Thus, two-photon absorption of 480 nm light can be used to excite a 240 nm absorbing fluorophore without the necessity of using ultraviolet sources. For two or more photons to be absorbed at the same time requires an extremely high photon density in the neighbourhood of the absorbing centre. This can only be achieved if an intense laser beam is focused into a small region using, in two-photon microscopy, the objective lens of the microscope (Figure 9.35). Thus, the image obtained is only from fluorescent centres that are in focus, but only a very small field is available. To offset this, the laser beam is scanned over the sample and the image recorded digitally. As with all fluorescence microscopy, scattering is a problem and can reduce image quality.

9.12 Fluorescent Nanoparticles

The subject ﬁeld covered by the term nanoparticles is enormous. In this section, attention is focused on the organic and inorganic materials described earlier in this chapter. These materials are often nontoxic compared Colour and the Optical Properties of Materials 412

microscope objective lens

cover glass

sample laser beam focal plane

fluorescent centres

glass slide Figure 9.35 Two-photon fluorescence microscopy: the fluorescence only occurs where the intensity of the incident beam is high, generally in the focal plane of the microscope objective. This means that all the fluorescing centres are in focus

with semiconductor quantum dots prepared from CdS and related chalcogenides (see Section 10.10) and so are preferred for many tagging purposes. Broadly speaking, nanoparticles of inorganic phosphors have fluorescence spectra similar to those of bulk materials. However, the large relative surface area and frequently lower crystallinity of these samples mean that the quantum efficiency of a nanoparticle cluster is usually lower than that of a similar bulk sample. This is due to energy transfer to the surface, surface defects and surface quenching of ions. This shortcoming can be ameliorated by enveloping the nanoparticles in a suitable shell of a similar material; for example, fluorescent lanthanoid-doped CePO4 nanoparticles can be given a shell of LaPO4. Many of the surface problems are now suppressed and the quantum efficiency of such core shell composites can be high. An alternative approach is to precipitate nanoparticles within a solid matrix, such as a glass, by, for example, heating or laser irradiation. The fluorescence from these inclusions is only rarely influenced by the surrounding solid. In the case of fluorescent molecules, the active phase can be coated onto the exterior of an inert nanoparticle such as silica (SiO2). In this case, although the surface effects are not removed completely, the fluorescent efficiency of the molecules can be adequate for many purposes, such as in the study of living cells. The small dimensions of nanoparticles mean that they do not scatter light. If such particles are embedded in transparent materials, the result is a clear composite. This opens the possibility of making transparent and fluorescent thin films or coatings on a wide range of substrates. Such films have found applications as sensors.

9.13 Fluorescent Markers and Sensors

Fluorescent markers are commonplace. Many banknotes, passports and other security documents are treated with ﬂuorescent dyes that are incorporated in printing inks. The ﬂuorescent markings need to be invisible under normal circumstances, which means that the molecules should not be strongly coloured or at least be masked by another dye molecule so as to be rendered undetectable. These invisible markings become bright when illuminated with ultraviolet light. Sensors to detect molecules need to be more sophisticated, and there are a number of ways in which these operate. Broadly speaking, the sensor can change in one of two ways. (i) The intensity of 413 Luminescence

fluorescent emission can vary as a function of the concentration of the analyte. This may simply be an effect related to fluorescence quenching or be caused by more complex interactions. (ii) The position of the emission band can change as a function of the concentration of the analyte. This is related to the solvatochromic effect described above and can occur if a degree of bonding occurs between the analyte molecules and the fluorophore. An ideal response is one in which a change in fluorescence is sharp, in which case the sensor may display a digital (on/off) output. An example of this digital-type response is provided by a pH sensor that switches from nonfluorescent to fluorescent as the pH passes a required value. The sensor consists of a pH-sensitive polymer containing a water-sensitive fluorophore. Typically, the polymer adopts an open hydrated form in low pH (i.e. H þ -rich) environments. The solvent has access to the fluorophore, interacts with it and successfully quenches emission. At high pH (i.e. low H þ ) environments, the polymer dehydrates and contracts upon itself into a globular structure. The solvent does not have access to the fluorophore, which is able to fluoresce under the appropriate excitation. A careful tailoring of the polymer structure and the fluorophore chosen allows this on/off effect to be sharp and to be tuned to operate over a range of specific pH values. That is, the fluorescent molecules light up at a specific pH. A similar method has been used to measure temperatures inside living cells. In this instance, the polymer that carries the water-sensitive fluorophore is heat sensitive rather than pH sensitive. At lower temperatures (with respect to normal metabolism) the polymer adopts an expanded and open state. The fluorophore is able to interact with the watery cell contents, allowing a degree of quenching that results, at best, in weak fluorescence. As the temperature increases, the polymer contracts and the fluorophore molecules become increasingly protected, resulting in an increase in fluorescent emission output. The intensity of the emission then acts as a temperature indicator. As before, the temperature range and colour of the emission can be tuned by varying the polymer and fluorophore. The detection of explosives is an important area of research where fluorescence can be helpful. An example is given by a sensor for the volatile components of explosives, especially TNT (2,4,6-trinitrotoluene). Here, the sensor is laid onto a glass surface as a thin film. The fluorescence is excited at 370 nm and the emission is at 408 nm. Once again, detection of the explosive relies upon emission quenching in the presence of the TNT vapour. Clearly, for such a sensor to be effective, it is important that other common molecules, such as perfume or fruit odours, do not interfere with the operation of the fluorophore. In addition, it must be reversible. That is, once the explosive is removed, the sensor must return to its initial state and show fluorescence again. Details of these and other fluorescent sensors are given in Further Reading.

9.14 Chemiluminescence and Bioluminescence

Chemiluminescence is light emitted as a result of a chemical reaction that leaves product molecules in a high- energy state from whence they return to the ground state by the emission of photons. The best known chemiluminescent reactions are associated with glow sticks. A glow stick consists of a transparent plastic tube containing one of the active chemicals and a fragile glass ampoule or inner tube containing the other reactant (Figure 9.36). To activate the glow stick, the outer tube is twisted or bent in order to fracture the inner glass tube, thus allowing the chemicals to mix. The resulting chemical reaction excites incorporated dye molecules which, in turn, give out light. The energy-providing chemical is hydrogen peroxide (H2O2), contained in the outer part of the stick. The glass tube contains diphenyl oxalate and the chosen dye. Hydrogen peroxide oxidizes the diphenyl oxalate to phenol and the very unstable intermediate 1,2-dioxetanedione, which decomposes immediately to carbon Colour and the Optical Properties of Materials 414

transparent plastic outer case

H2O2 solution diphenyl oxalate + dye solution in thin-walled glass tube

Figure 9.36 Glow stick (schematic). The inner thin-walled glass tube is broken when the flexible outer plastic tube is bent, allowing the reactive chemicals to mix and generate light by chemiluminescence dioxide, exciting the dye molecules in the process (Scheme 9.1). The concentrations of the chemicals and the temperature influence the length of time over which the glow stick is luminous. A large number of dyes have been used in glow sticks. Two of the commonest are rubrene (5,6,11,12- tetraphenylnaphthacene), which gives a yellow orange fluorescence, and 9,10-diphenylanthracene, which gives blue (Figure 9.37). Bioluminescence is a form of chemiluminescence in which light is emitted by living organisms. Bioluminescence occurs widely, from bacteria, single-celled algae, many marine organisms, cnidarians (jellyfish and relatives) to insects. The energy emission from the excited molecules can be rapid, in biofluorescence, or delayed, in which case it is correctly referred to as biophosphorescence. Bioluminescence is typified by insects such as the European glow-worm Lampyris noctiluca, which emits green light, and the North American firefly, Photinus pyralis, which has a yellow emission. The light-emitting organs, often called lanterns, differ in size and disposition from species to species and often from one sex to the other. For example, the wingless female of L. noctiluca has two large lanterns and four smaller ones along the sides of the abdomen, while the winged males have just two small lanterns at the tip of the abdomen.

O O (a)

H22 O + O O diphenyl oxalate

O O OH 2 +

O O phenol 1,2-dioxetanedione

(b) O O

+ dye 2 CO2 + dye*

O O 1,2-dioxetanedione

(c) dye* dye + light Scheme 9.1 The reactions taking place in glow sticks: (a) hydrogen peroxide and diphenyl oxalate form phenol and 1,2-dioxetenedione; (b) 1,2-dioxetanedione decomposes to carbon dioxide and the energy released is passed to the dye molecule to produce an excited state dye; (c) the excited dye molecule releases the energy in the form of light 415 Luminescence

(a) (b)

Figure 9.37 The idealized structures of (a) rubrene (5,6,11,12-tetraphenylnaphthacene) and (b) 9,10- diphenylanthracene

Because of the large numbers of organisms that display bioluminescence, there are a wide variety of adaptive purposes that light production is put to, including mate attraction and prey attraction. There are similarly a number of different light-producing mechanisms involved. It is possible, with a certain loss of precision, to generalize the light-producing mechanisms into two main pathways. In beetles such as glow-worms, the reaction usually involves a molecule of a light-producing chemical generally called a luciferin which, in the presence of oxygen, the energy providing molecule ATP (adenosine triphosphate) and a catalytic enzyme, a luciferase, produces an unstable dioxetanone intermediate (similar to those produced in glow sticks), which decomposes spontaneously to an excited oxidized oxyluciferin, which loses energy by photon emission (Scheme 9.2a). The processes involved in this last step frequently generate light by transitions from an upper p state to a lower p energy level. Luciferin is not a single compound, and luciferins and luciferases differ from one species to another, thus accounting (at least in part) for the different colours produced. One of the most widely studied is luciferin extracted from the North American firefly P. pyralis. It is optically active and only one enantiomer is biochemically active in light production. It has the (daunting) name 2,4-dihydro-2-(6-hydroxy-2-benzothia- zolyl)-4-thiazolecarboxylic acid (Scheme 9.2b). In this insect it appears that the flashing is controlled by the production of nitric oxide (NO) in the lantern. This reactive gas is an oxygen scavenger. It is believed that NO formation inhibits the consumption of oxygen for respiration, allowing it to be used in the light-producing reaction via the oxidation of luciferin. A different strategy is used in the coelenterate medusa (jellyfish) A. victoria, famous as the original source of GFP (Section 9.11). In this and similar species, the luciferin, coelenterazine, is combined with the luciferase

→ → (a) luciferin + O2 + luciferase oxyluciferin* oxyluciferin + light (excited) (ground state)

(b) O

HO S N C OH

N S Scheme 9.2 (a) The generalized reaction sequence of light production in the North American firefly Photinus pyralis; (b) the structure of the North American firefly luciferin Colour and the Optical Properties of Materials 416 catalyst and oxygen into a single photoprotein called aequorin. In the presence of a trigger, in this case Ca2 þ ions, the luciferin, if isolated, emits blue light of wavelength approximately 470 nm. However, in the living animal, the luciferin is in close proximity to a molecule of GFP. Energy transfer occurs and the complex emits green fluorescence at 509 nm instead of blue light. Because of the vast range of bioluminescent species, it is certain that other variations on these themes will be discovered in the future.

9.15 Triboluminescence

This is the property of a material to emit light on crushing or scratching. It is easily seen when sugar is ground in a darkened room or when a sugar-rich sweet is bitten in two. Besides sugar, many minerals, including calcite, various feldspars, fluorite and sphalerite, show the phenomenon. It is also seen if a strip of tape is rapidly pulled from a roll of adhesive tape in the dark. The source of the light emission in these cases is often not clear and a number of mechanisms have been proposed to account for it. One set of explanations centres upon the fact that the release of mechanical energy at a crack tip over small time scales can create very high temperatures, sufficient to raise the surface to incandescence. In the case of metals, oxidation of the newly formed surface also contributes greatly to the temperature and can enhance light emission. For example, metallic glasses with compositions near to Zr41.2Ti13.8Cu12.5Ni10Be22.5 emit a broad band of intense light equivalent to a black body temperature of about 3175 K when fractured in air. It is surmised that the fracture exposes fresh metallic surfaces and that oxidation of these, a strongly exothermic process, causes the increase in temperature and the emission of light. The rupture of chemical bonds at the growing tip gives rise to unpaired electrons. These can react with gaseous molecules, providing a further source of light emission. In insulating materials, the separation of the surfaces can also create considerable electric fields. The recombination of separated charges may give out light. Alternatively, the field created might be of sufficient strength to excite gases such as nitrogen to such an extent that they can emit ultraviolet radiation. In fact, it is believed that when sugar is fractured the blue glow is due to the visible tail of the ultraviolet emission from excited nitrogen molecules. Ultraviolet light generated in this way can be absorbed by a fluorophore, should any be present, and also emit visible light. There is some interest in using triboluminescence in sensors to detect fracture damage that may occur; for instance, when a bird strikes an aircraft. Strongly triboluminescent crystals are embedded in resin and applied to the structure to be monitored. Upon fracture, the triboluminescent crystals emit a brief flash of light that can be collected by an optical-fibre cable and transferred to a recording instrument. The intensity of the light gives an indication of the severity of the fracture damage. In addition, a variety of materials that give out different colours can be distributed over the component to be monitored, so that the location of the damage becomes readily apparent. The threshold at which damage can be detected is a function of the type of triboluminescent crystals used, and the density and particle size of crystals contained in the resin. This refinement allows trivial impacts to be discounted.

9.16 Scintillators

Scintillators are luminescent materials that emit light in response to the impact of heavy particles or highly energetic radiation. They are used for the detection of electrons, neutrons, a-particles, X-rays, g-rays and so on. Applications range from security scanners, industrial inspection units, medical diagnostic imaging and high-energy physics. 417 Luminescence

Considering that these applications span many orders of magnitude of energy and particle characteristics, it is not surprising that a large number of different scintillator materials have been tested. These include inorganic crystals, polycrystalline ceramics and glasses, organic plastics and liquids, and inert gases. Despite this variety, there are a number of properties that these materials must have in common. These are: 1. Transparency. The method of detection of the high-energy radiation is by visible light emission, so that the scintillator must be transparent at the appropriate wavelengths. 2. High attenuation. Clearly, the stopping power of the scintillators in important. If this is low, then many of the particles to be detected will pass straight through the detector without giving a signal. 3. High light output. The output power must be sufficiently high that photodetectors record every event. 3 þ The neutron scintillator Cs2LiYCl6 doped with Ce is able to emit 70 000 photons per neutron absorbed. 4. Decay time. A high intrinsic decay time is important for many particle physics applications, as each event needs to be recorded separately. However, in imaging applications, longer decay times are often necessary to give a brighter image. 5. Low afterglow. As with the decay time, significant afterglow can interfere with the recording of single events. However, it may be useful in some imaging equipment in contributing to a brighter image. 6. High threshold for radiation damage. It is obvious that, as the radiation to be detected is highly energetic, the detector must be able to withstand considerable exposure. Liquid scintillation counters are widely used to record radioactivity from b-emitters. These materials are generally unstable radioactive isotopes used for medical diagnostics, the b-radiation consisting of energetic electrons expelled from the radioactive nuclei. The liquid in the counter is frequently benzene or toluene. These organic molecules absorb the energy of the b-rays and are excited to higher energy levels. Usually, they do not emit light directly, but transfer energy to a fluorescent dye molecule dissolved in the liquid medium and this in turn emits photons. X-ray tomography uses ceramic specimens of (Y,Gd)2O3 doped with Eu, or Gd2O2S doped with a mixture þ of PrF3 and CeF2, single crystal caesium iodide (CsI) doped with Tl (CsI:Tl), as well as some of the single- crystal detectors listed below. X-ray detectors for other purposes include a mixed barium halide BaFBr1 xIx 2 þ doped with Eu , (CsI:Tl) and calcium tungstate CaWO4. In the systems containing a lanthanoid the detectors fluoresce in the same way as the lanthanoid-containing luminescent materials described above. That is, the energy from the X-rays excites an electron into a high energy band. Thereafter, radiationless decay allows the energy to degrade until an f energy level on the dopant activator is reached. A photon is subsequently emitted to lower the energy to either the groundstateorclosetoitbyanfftransition.Inthe 2 case of CaWO4, the energy input results in charge transfer within the WO4 units. The return to the ground state is by photon emission. Positron emission tomography uses radioactive nuclei that emit positrons as the radiation source. Positrons are positive electrons, and in ordinary matter they are short lived. Each positron soon collides with an electron. Both particles are eliminated and give rise to two energetic photons. The detectors employed are mainly single-crystal sodium iodide (NaI) doped with thallium iodide (NaI:Tl), CsI:Tl or single crystals of the oxide Bi4Ge3O12. The Tl-doped crystals emit a green light of wavelength close to 550 nm from transitions between the Tl energy levels. High-energy physics uses a wide variety of mainly single-crystal detectors. These include NaI:Tl, CsI:Tl and lead tungstate (PbWO4). This latter material is used in high-energy particle detectors in the recently commissioned Large Hadron Collider at CERN. A charge transfer process, similar to that in CaWO4,is responsible for light emission. Colour and the Optical Properties of Materials 418

Further Reading

The Kroger Vink notation for defects is explained in R. J. D. Tilley, Defects in Solids, John Wiley and Sons, Inc., Hoboken, NJ, 2008. Material relevant to this chapter is contained in several articles in J. P. Hornak (ed.), Encyclopedia of Imaging Science and Technology, John Wiley and Sons, Inc., New York, 2002, including: Optical microscopy, p. 1106; Laser fluorescence imaging, p. 861; Cathode ray tubes, p. 44; X-ray fluorescence microscopy, p. 1475. The luminescent phosphors used in fluorescent lamps, CRTs and other devices are described by G. Blasse, B. C. Grabmaier, Luminescent Materials, Springer-Verlag, Berlin, 1994. T. Justel,€ H. Nikol, C. Ronda, Angew. Chem. Int. Ed. 37, 3084 3103 (1998). H. A. Hoppe,€ Angew. Chem. Int. Ed. 48, 3572 3582 (2009). A survey of competing flat-panel display types with emphasis on plasma displays is given in A. Sobel, Sci. Am. 278 (May), 48 55 (1998). More technical information on displays is given in Mater. Res. Soc. Bull., 23 (March), (1996), which includes: Y. Yang, Polymer electroluminescent devices, p. 31. J. Hanna, I. Shimizu, Active matrix liquid crystal displays, p. 35. T. Tsutsui, Molecular thin films, p. 39. P. D. Rack, A. Naman, P. H. Holloway, S-S. Sun, R. T. Tuenge, Electroluminescent displays, p. 49. L. F. Weber, J. D. Birk, Colour plasma displays, p. 65. For an introduction to protein chemistry, see D. Whitford, Proteins, Structure and Function, John Wiley and Sons, Ltd, Chichester, 2005. GFP and related matters are reviewed by O. Shimomura, Angew. Chem. Int. Ed. 48, 5590 5602 (2009). M. Chalfie, Angew. Chem. Int. Ed. 48, 5603 5611 (2009). R. Y. Tsien, Angew. Chem. Int. Ed. 48, 5612 5626 (2009). Much relevant up-to-date information on fluorescence-related microscope techniques is to be found on the websites of microscope manufacturers, including Olympus, Nikon and Zeiss. The experiments on DNA uncoiling are in J. van Marneren, P.Gross, G. Farge, P.Hooijman et al., Proc. Nat. Acad. Sci. U. S. A. 106, 18231 18236 (2009). Triboluminescence is discussed by C. J. Gilbert, J. W. Ager, V.Schroeder, R. O. Ritchie, J. P.Lloyd, J. R. Graham, Appl. Phys. Lett. 74, 3809 3811 (1999). I. Sage, Chem.Br. (February), 24 27 (2001). Information on scintillators is given by C. Greskovich, S. Duclos, Annu. Rev. Mater. Sci. 27, 69 88 (1997). http://scintillator,lbl.gov/. 10 Colour in Metals, Semiconductors and Insulators

. How can colourless boron impurities tint diamond blue? . What produces light in a light-emitting diode (LED)? . Why are copper and gold coloured, whereas most metals resemble silver?

So far, the broad scheme of bonding in solids has been ignored. This chapter addresses this lack, and the colours and optical properties that arise as a consequence of this bonding are described. In order to achieve this, it is necessary to know something of theway that the outer electrons on the component atoms of the material are held in metals, semiconductors and insulators. This is described by band theory. In this approach, the outer electron energy levels are shown to be broadened into energy bands as the atoms coalesce into a molecule and then a solid. The main energy landscape in a solid is then the energy band structure. This process can be viewed as an extension of the ideas that delocalize atomic orbitals into molecular orbitals, but now these orbitals extend throughout the solid rather than just over the molecule. Thus, transitions in an atom, between sharp energy levels, change into transitions between a HOMO and a LUMO in a molecule and then into transitions between a lower energy valence band and a higher energy conduction band in a solid. In the simplest depictions, the upper energy band (the conduction band) is separated from the lower energy band (the valence band) by a constant band gap. This is called the ﬂat band model. In real structures, the band architecture is complex. Note that these concepts are reversible. The properties of a bulk solid will change as the degree of division of the solid increases until, at the smallest dimensions, the properties become less and less referable to energy bands identical to those of the bulk, and must be considered in terms of molecular and then atomic energy levels. These changes come to the fore in nanoparticles, described in more detail below.

Colour and the Optical Properties of Materials Richard J. D. Tilley Ó 2011 John Wiley & Sons, Ltd Colour and the Optical Properties of Materials 420

It is most convenient to start with a consideration of insulators, as these provide links with the previous chapters.

10.1 The Colours of Insulators

Insulators have the upper energy band completely empty and the lower energy band completely filled by electrons (Figure 10.1a). The filled energy band is called the valence band and the empty energy band is called the conduction band. The energy difference between the top of the valence band and the bottom of the conduction band is the band gap, magnitude Eg. A typical insulator is characterized by a large band gap (Table 10.1). If light falls onto an insulator, it will not be absorbed unless the energy of the incident photons is high enough to promote an electron from the valence band to the conduction band. The photon energy at this point is a measure of the optical band gap. In a flat band model (i.e. Figure 10.1a), this is a single energy and a sharp step in the absorption spectrum would be expected, called the band edge or the absorption edge (Figure 10.1b). In real solids the band gap is of more complex geometry, and the transition is not so sharp in practice. This means that there is an uncertainty involved in the estimation of the band gap from spectra, and a range of values are found in the literature. (Other techniques are also used to measure the band gap of solids, and these also give values slightly different from those obtained from spectra, adding to the spread of recorded values.) In most insulators, oxides for example, the optical band gap lies in the ultraviolet part of the spectrum. Thus, although there is very strong absorption at these wavelengths, the visible spectrum is not affected. Crystals of insulators are then transparent and powders are white, due to scattering and reflection, as detailed in earlier chapters. However, the absorption due to thevalence band conduction band transition has a certain width. If the optical band gap energy falls just into the ultraviolet, roughly speaking below 3.1 eV, the band gap absorption spectrum impinges into the violet end of the spectrum. This tends to give oxides a yellow tint, typified by lemon yellow tungsten trioxide (WO3; Figure 10.1c) and pale yellow ceria (CeO2; Section 7.15). The fact that many insulators strongly absorb in the ultraviolet but are transparent to visible radiation is made use of in sunscreens (Section 5.7). In particular, ZnO and TiO2 are very widely applied. These absorb the incident harmful ultraviolet radiation and, provided that the particle size is sufficiently small, are invisible to the eye. The optical band gap of a solid varies with particle size. Although this has little consequence for ordinary fine powders and polycrystalline thin films, such as those used in paints and sunscreens, a change is observed at the smallest particle sizes. Nanoparticles show a considerable shift in band gap energies, with the band gap increasing as the particle size drops below approximately 10 nm diameter (Figure 10.2). The optical band gap of many materials has also been found to decrease slightly with temperature. Although this effect is small, it can lead to interesting colour changes. White zinc oxide (ZnO) absorbs in the near ultraviolet. At high temperatures the decrease in the band gap means that some violet light is absorbed. The material will then become yellow to the eye. This effect is also noticeable with the yellow oxide In2O3. At room temperature this absorbs in the green blue. As the temperature increases, the absorption shifts towards the lower energy red, causing the oxide to take on a much deeper yellow brown colour. The effect of colour variation with temperature is known as thermochromism. This term was encountered in Section 6.9. with respect to liquid-crystal thermometers.1

1 Note that the name thermochromism applies to the change of colour with temperature. That is, it does not describe the mechanism, only the observed effect. The two examples mentioned have quite different mechanisms. Other mechanisms for various thermochromic changes are also found, especially among organic thermochromic materials. 421 Colour in Metals, Semiconductors and Insulators

conduction band (empty)

100% band gap, energy Eg Absorption

0 valence band (full) (a) (b) Eg Energy

100 ZnO

WO3

50 Reflectance %

400 500 600 700 800 (c) Wavelength / nm

Figure 10.1 (a) Simple ‘ﬂat band’ approximation of the band structure of an insulator. (b) The absorption of energy by an insulator (schematic). (c) The reﬂectance spectra of white zinc oxide (ZnO) and pale yellow tungsten trioxide (WO3)

10.2 Excitons

One of the most important aspects of the band theory of solids is that, when an electron is promoted from the valence band to the conduction band, two free charged ‘particles’ form: an electron now in the conduction band and an electron hole, more commonlyjust called ahole, in the valence band. Holes contributesigniﬁcantly to the electronic properties of the solid and can be somewhat loosely considered to behave as if they were Colour and the Optical Properties of Materials 422

a Table 10.1 Optical band gap Eg of bulk oxides

Oxide Band gap/eV Oxide Band gap/eV

MgO 8.7 MgAl2O4 7.5 Al2O3 6.3 SrZrO3 5.4 Y2O3 5.8 La2Ti2O7 4.0 Hf2O3 5.2 LiNbO3 3.8 Ga2O3 4.8 LiTaO3 3.8 ZrO2 4.6 NaTaO3 3.8 Ta2O5 3.9 MgTiO3 3.7 SnO2 3.5 Al2TiO5 3.6 Nb2O5 3.3 KNbO3 3.3 TiO2 3.2 BaTiO3 3.2 ZnO 3.3 SrTiO3 3.1 Sb2O3 3.0 In2O3 2.8 Bi2O3 2.8 WO3 2.7

a Data from F. Di Quarto, C. Sunseri, S. Piazza, M. C. Romano, J. Phys. Chem. B101, 2519–2525 (1997). For alternative data, see J. Portier et al., Prog. Solid State Chem., 32, 207–217 (2004).

3.9

3.8

3.7

3.6

3.5

Band gap / eV 3.4 bulk 3.3

3.2 1234567 Particle size / nm Figure 10.2 Variation of the band gap of ZnO nanoparticles as a function of particle size. [Data from R. Viswanathan et al., J. Mater. Chem., 14, 661–668 (2004); H.-M. Xiong et al., Angew. Chem. Int. Ed., 48, 2727–2731 (2009)]

‘positive electrons’.2 Because these two species have opposite charges, they attract via Coulomb forces. The bound pair is called an exciton. The energy required to create an exciton is equal to the energy required to promote the electron into the conduction band, the band gap energy Eg, minus the binding energy of the exciton. This is depicted as a new energy level just below the conduction band (Figure 10.3a). The binding energy

2 These are not genuine positive electrons (positrons), because such a particle would be eliminated instantaneously by combination with a normal electron. They are virtual particles equivalent to the absence of an electron. 423 Colour in Metals, Semiconductors and Insulators

conduction band

exciton energy levels

valence band (a)

exciton peaks Absorption

(b) Eg Energy

Figure 10.3 Exciton energy levels: (a) energy levels just below the conduction band in a semiconductor; (b) exciton absorption peaks close to the absorption edge in a semiconductor depends upon a number of factors, and there may be several closely spaced levels present rather than one. The presence of excitons in a crystal will then be revealed by absorption peaks corresponding to transitions between the valence band and the exciton energy levels which lie on the low energy side of the absorption edge (Figure 10.3b). Generally, this interaction energy is weak in insulating (and semiconductor; Section 10.6) crystals. The electron and hole are not close and the exciton is considered to extend over several unit cells of the structure. Excitons representing this situation are called Mott Wannier excitons or free excitons. A free exciton can be thought of as analogous to an extended hydrogen atom with a hole replacing a proton and the energy levels that lie below the absorption edge analogous to hydrogen atom energy levels. The idea of considering an exciton as an ‘atom’ is, in fact, close to the original conception of an exciton. In the 1930s, Frenkel suggested that some aspects of the ultraviolet absorption spectra of insulators could be explained if some atoms were in a state of excitation; that is, were excitons. In this case, the excited electron of the electron hole pair is in an upper atomic orbital rather than the conduction band of a solid and the hole is in a lower atomic orbital rather than the valence band of a solid. The hole and electron remain in close proximity, within the atomic orbital structure of the excited atom, and the hole electron interaction energy is high. Such excitons are called Frenkel excitons or tightly bound excitons. Frenkel excitons may occur in insulating solids such as Colour and the Optical Properties of Materials 424 the ionic oxides and halides. For example, in the alkali-metal ﬂuorides, where excitons are located on anions, an electron is promoted from one of the ground-state (1s2 2s2 2p6) orbitals to one of the higher (3s, 3p, ...) empty ones. The exciton energy levels will now take values close to those of the ionic energy levels of the F ion. Excitons can also form in molecular crystals, such as those of the dye molecules and conjugated or aromatic hydrocarbons. In these cases the bonding within the molecule is strong and represented by a series of molecular orbitals. Bonding between the molecules is rather weak. Excitons will then be associated with a hole in a low-energy molecular orbital, typically a HOMO, and an electron in a higher energy molecular orbital, typically a LUMO. Exciton energy levels will then be similar to the molecular orbital energy ladder. The concept of an exciton, therefore, spans the range from a strongly bound electron hole pair on an atom to a weakly bound pair of virtually free particles moving through the band structure of the solid. In all cases, the excitons are revealed by extra absorption peaks in the spectrum of the material. However, these are mostly observed when the sample is at low temperatures, as thermal vibrations smear out the absorption peaks at normal temperatures.

10.3 Impurity Colours in Insulators

Impurities are the commonest way to introduce colour into colourless insulators. These impurities are usually regarded as defects; the commonest of these are point defects, which are located at a particular atom site within the solid matrix. ‘Coloured’ transition metal or lanthanoid ion impurities in glasses, gemstones and phosphors, which have already been described, fall into this category (Chapters 7 and 9). In these compounds, the impurity atom or ion occupies a position normally ﬁlled by one of the component atoms of the structure, such as impurity Cr3þ in an Al3þ site in ruby. The energies of the colour-producing optical transitions, d d or f f transitions, are much lower than the band gap (2 eV compared with 6.3 eV for ruby) and the energy levels giving rise to colour lie in the band gap between the valence and conduction band of the solid. In the case of lanthanoid phosphors, exciting radiation is often energetic enough to promote an electron from the impurity atom into the conduction band originating in the outer electron orbitals of the surrounding matrix atoms. Energy is lost by nonradiative transitions until an energy level located on the lanthanoid ion is reached, after which a visible photon is released (Figure 10.4). New energy levels can also be introduced into the band gap by the addition of ‘colourless’ impurity atoms and other point defects, including the absence of an atom from a normally occupied site a vacancy. The impurities are classed as donors if they normally contribute electrons to the conduction band or as acceptors if they normally take up electrons from the valence band. Donor dopants may give energy levels close to the conduction band or far from it. Similarly, acceptor dopants may give energy levels close to the valence band or far from it. Those close to the band edges are called shallow levels, while those towards the centre of the band gap are called deep levels. The excitation of electrons to and from these levels will give rise to colours when the energy difference falls in the visible range (Figure 10.5).

10.4 Impurity Colours in Diamond

The processes leading to colour when ‘colourless’ dopants are introduced into a transparent insulator are well illustrated by the coloration induced in diamonds, which varies from brownish (low value) coloration through orange, pink and purple to the highly prized and rare blue and yellow gems. The diamond structure is built up of carbon atoms each surrounded by four carbon atom neighbours in a tetrahedron, the linking being via sp3 hybrid bonds (Figure 10.6a). Diamond has a band gap (of about 5.5 eV) 425 Colour in Metals, Semiconductors and Insulators

conduction band

lanthanoid transition metal impurity impurity

valence band

Figure 10.4 Absorption and emission of radiation by transition and lanthanoid metal ions in an insulator. The energy levels important for colour are situated in the band gap of the insulator. Efﬁcient ﬂuorescence often uses excitation to the conduction band as a preliminary to the emission of visible light. [Nonradiative transitions are shown as dotted lines] which is too large to absorb visible light; therefore, perfect diamonds are clear. The commonest impurity in natural diamonds is nitrogen (N). Most of these nitrogen atoms substitute for carbon on normal tetrahedral sites in the crystal. Diamonds are often subjected to temperatures of 1000 1200 C over geological timescales, allowing these nitrogen atoms to diffuse through the structure, and most end up in clusters, some of which produce a straw-yellow colour. Such stones are known as Cape yellows and are of considerable value in jewellery. More rarely, some diamonds contain nitrogen as isolated N atoms located on carbon sites; this group also includes yellow diamonds (canaries) that are similarly highly prized.

conduction band

donor levels

acceptor levels

valence band

Figure 10.5 Schematic band structure of an insulator containing defects that introduce additional energy levels in the band gap. Any transitions that are of a suitable energy can cause the solid to become coloured Colour and the Optical Properties of Materials 426

a (a) (b) C

N conduction electron band

~2.2 eV ′ NC

~5.5 eV

valence band (c)

Figure 10.6 (a) The structure of diamond. Each carbon atom is surrounded by four others at the vertices of a tetrahedron. (b) Idealized representation (ignoring atomic relaxation and molecular orbital formation) of a single 0 0 nitrogen substitutional impurity NC . (c) Energy levels of a substitutional N impurity NC

The colour of these latter diamonds is caused by the isolated nitrogen impurities in the following way. Nitrogen, with an electron configuration 1s2 2s2 2p3, has five bonding electrons, one more than carbon, with configuration 1s2 2s2 2p2. Substitution of nitrogen for carbon on a normal carbon atom site in the crystal creates 0 3 an NC defect with an effective negative charge (Figure 10.6b). Four of the electrons around each impurity nitrogen atom are used to fulfil the local sp3 bonding requirements of the crystal structure and one electron remains unused. The extra electron, one per nitrogen atom impurity, can be excited by suitable radiation into the conduction band and so the defect is a donor impurity. On an energy-level diagram this is often represented as a deep donor level in the energy gap centred at approximately 2.2 eV below the conduction band, but because of lattice vibrations and other interactions it is better regarded as a narrow band of energies centred at 2.2 eV and extending to 1.7 eV, the quoted ionization energy of the nitrogen atom in diamond (Figure 10.6c). This centre is able to absorb all visible light of wavelengths longer than about 564 nm, giving the stones a faint yellow aspect. As the nitrogen concentration increases, the colour intensifies.

3 The nomenclature is that of the standard Kroger Vink notation (see this chapter’s Further Reading). 427 Colour in Metals, Semiconductors and Insulators

The most common impurity in diamond appears to consist of a pair of nitrogen atoms on adjacent carbon- 20 atom sites (NC NC) . These form deep levels at approximately 4 eV below the conduction band. These centres absorb 310 nm radiation and do not contribute to the colour of stones. From among the many other nitrogen-containing clusters known, the N3 centre, which consists of three nitrogen atoms on adjacent carbon sites in a planar fashion around a carbon vacancy (i.e. a missing carbon . atom), ðNC-- NC-- NC-- VCÞ , seems to be responsible (at least in part) for the pale straw colour of Cape yellow diamonds (Figure 10.7a). The N3 centre absorbs just in the blue end of the visible, at 415 nm, giving yellow stones. The N3 centres are often accompanied by N2 centres consisting of two nitrogen atoms on normal carbon sites adjacent to a carbon vacancy, i.e. (NC VC NC) clusters. These can be electronically neutral, in which case they absorb at approximately 475 nm, giving a yellow colour to the stones and adding to that contributed by the N3 clusters. They emit a strong green light at 531 nm after excitation with laser light. The N2

VC VC

(a) (b)

3E ~ 3 meV C N electron

1.945eV

Figure 10.7 Idealised representations (ignoring atomic relaxation and molecular orbital formation) of nitrogen- ð Þ. containing defect centres in diamond: (a) an N3 centre, NC-- N C-- N C-- V C ; (b) an N–V centre, (NC–VC ). (c) Energy levels of an N–V centre. (These energy levels fall within the band gap of diamond) Colour and the Optical Properties of Materials 428 cluster can also be negatively charged, (N V N)0, in which case the absorption is at approximately 989 nm in the infrared. This absorption band can spill over into the red part of the visible spectrum, leading to stones with a blue tone. When all these clusters are present in roughly equal quantities, a green colour is perceived. The cluster studied in most detail is that consisting of a single nitrogen impurity located next to a carbon vacancy, (NC VC), mostly written as (N V), as these endow diamond thin films with interesting electronic properties. These centres are readily created by the irradiation of artificial diamonds or diamond thin films, which normally incorporate nitrogen impurities during synthesis, with high-energy protons. The proton irradiation results in the formation of carbon vacancies, and if the crystals are then annealed at above 600 C, the temperature at which the vacancies become mobile, they diffuse through the structure until they encounter a nitrogen impurity. The strain around the nitrogen atom effectively traps the vacancy, preventing further migration. In the resultant (N V) centres, the tetrahedron surrounding the carbon vacancy is composed of three carbon atoms and one nitrogen atom (Figure 10.7b). These centres can be electronically neutral, but 0 the most studied cluster is the negatively charged (N V) centre [(NC VC) ]. These absorb strongly at approximately 575 nm, giving stones a pink hue. They are strongly photoluminescent, and excitation with laser light in the range from 490 to 560 nm results in a strong red emission peak with a maximum at approximately 670 nm. The electron trapped at the centre has an orbital that extends over the cavity, so that it encompasses not only the nitrogen impurity, but also the three carbon atoms that also surround the vacancy. The energy levels of this cluster can be calculated by molecular orbital theory. To a first approximation, the ground-state term is 3A and the first excited state is 3E (Figure 10.7c). However, spin orbit coupling (Section 7.2) splits the ground state and the excited state into two. These have different energies, and because of the spin-multiplicity rule, when an (N V) centre emits a photon, the transition is allowed from one of these and forbidden from the other. Moreover, the electron can be flipped from one state to another by using low-energy radio-frequency irradiation. Irradiation with an appropriate laser wavelength will excite the electron and as it returns to the ground state will emit fluorescent radiation. The intensity of the emitted photon beam will depend upon the spin state, which can be changed at will by radio-frequency input. In addition, the application of a magnetic or static electric field splits the levels again. The extent of this splitting depends upon the magnitude and relative orientation of the applied field and the defect. The transitions between the ground-state and excited-state levels are subject to different transition rules than those in the absence of these fields. Thus, the intensity of the output fluorescence may be modulated by the imposition of radio-frequency radiation, by magnetic fields and electrostatic fields. Unsurprisingly, these centres are under active exploration for use as components for the realization of quantum computers. Although nitrogen impurities give rise to the highly valued yellow-hued diamonds, other colourless impurities are also important. For example, prized blue diamonds are the result of boron impurities. In . this case, each boron impurity atom occupies a carbon position, again forming a substitutional defect, BC (Figure 10.8a). Boron, with an electron configuration 1s2 2s2 2p1, has only three outer bonding electrons instead of the four found on carbon. These three are used in fulfilling the bonding requirements of the structure, but one bond of the four is incomplete and lacks an electron, giving the defect an effective positive charge. In semiconductor physics terms, each boron atom dopant has an accompanying hole in proximity to the occupied site and is an acceptor impurity. This is represented by the creation of a set of new acceptor energy levels approximately 0.4 eV above the valence band (Figure 10.8b). The transition of an electron from the valence band to this acceptor level has an absorption peak in the infrared, but atomic vibrations and other imperfections broaden this into a narrow band of energies allowing the high-energy tail of the absorption band to encroach into the red at 700 nm. The boron-doped diamonds, therefore, absorb some red light and leave the gemstone with an overall blue colour. Other colourless ions, such as of hydrogen, sulfur and phosphorus, have also been introduced into diamonds, especially with a view to altering the electronic properties rather than colour. 429 Colour in Metals, Semiconductors and Insulators

hole (a)

conduction band (empty)

~ 0.4 eV

valence band (b) (full)

. Figure 10.8 (a) The idealised structure of a substitutional B atom in diamond BC. (b) Acceptor energy level of the defect

10.5 Colour Centres

In the 1920s and 1930s there was considerable interest in the fact that synthetic alkali halide crystals could be made intensely coloured in a number of ways, including irradiation by X-rays, electrolysis (with colour moving into the crystal from the cathode), or heating the crystals at high temperatures in the vapour of an alkali metal. The principal investigator, Pohl, in Germany, attributed the colour to the presence of Farbzentren (lit. colour centres). It is now well known that exposure of transparent solids, both glasses and crystals, to high-energy radiation frequently makes them coloured and the colour arises because the treatment has introduced defects into the material. The defects responsible for this are known as colour centres. Many different colour centres have now been characterised. (Note that there is a certain degree of imprecision in the literature, and colours caused by impurities, described above, are also sometimes said to be due to colour centres.)

10.5.1 The F centre

The ﬁrst colour centre to be characterised was the F centre, a term derived from Farbezentrum (colour centre), before it was clear that many different colour centres can form. F centres were ﬁrst produced by exposing alkali halide crystals to high-energy radiation such as X rays. This causes the crystals to become brightly coloured with fairly simple bell-shaped absorption spectra. The peak of the absorption curve lmax moves to higher wavelengths as both the alkali metal ion size and halide ion size increase (Table 10.2). Colour and the Optical Properties of Materials 430

Table 10.2 Alkali metal halide F centres

l a Compound Absorption wavelength max/nm Colour Lattice parameter/nm b LiF 235, UV colourless 0.4073 NaF 345, UV colourless 0.4620 KF 460, blue yellow brown 0.5347 RbF 510, green magenta 0.5640 LiCl 390, UV (just) yellow green 0.5130 NaCl 460, blue yellow brown 0.5641 KCl 565, green violet 0.6293 RbCl 620, orange blue green 0.6581 LiBr 460, blue yellow brown 0.5501 NaBr 540, green purple 0.5973 KBr 620, orange blue green 0.6600 RbBr 690, red blue green 0.6854 a The appearance of the colour centre-containing crystal is the complementary colour to that removed by the absorption band. b UV ¼ ultraviolet.

F centres can be introduced in several ways, apart from using ionising radiation. One of these involves heating the crystals at high temperatures in the vapour of an alkali metal. It is notable that the exact metal does not matter as long as it is an alkali metal. That is, if a crystal of potassium chloride (KCl) is heated in an atmosphere of sodium vapour, typical violet KCl F centres are formed, not the orange brown NaCl colour centres. Another way of introducing F centres into alkali halide crystals is to pass an electric current through heated samples and electrolyse them. In this case, the typical F centre colour is seen to move into the crystal from the cathode region. Once again, the colour depends upon the crystal being electrolysed, not on the exact nature of the cathode. Thus, F centres in sodium chloride (NaCl) always give the crystal an orange brown colour irrespective of the method of generation. These observations suggest that the centres are defects inthe crystal structure that do not involve the chemical nature of the components of the material in a direct fashion. This is so, and it has long been known that the F centre is an anionvacancy plus a trapped electron (Figure 10.9). The trapping is due to the fact that the missing anion creates a vacancy that has an effective positive charge and it is this charge that attracts the electron to form . 0 a ðVXe Þ centre where X represents the missing anion. The F centre in its ground state forms a deep level in the band gap of the alkali halide solid. The electron in this location behaves rather like the electron surrounding a hydrogen atom, and is able to absorb electromagnetic radiation, causing it to be promoted from one energy level to another. These transitions give rise to the colour of the solid. If enough energy is supplied then the electron is promoted into the conduction band, where it is no longer trapped.

10.5.2 Electron and hole centres

Since the original studies of F centres, many other colour centres have been characterised which may be associated with either trapped electrons or trapped holes. These are called electron-excess centres when electrons are trapped and hole-excess centres when holes are trapped. The F centre is an electron-excess centre and arises because the crystal contains a small excess of metal. Similar metal-excess F centres exist in compounds other than the alkali halides. An example is provided by the 4 mineral Blue John. This is a rare, naturally occurring form of ﬂuorite (CaF2). The coloration is caused by

4 The name ‘Blue John’ is a corruption of the French term ‘bleu jeune’ which was used to describe the blue form of the normally yellowish ﬂuorite crystals found in nature. 431 Colour in Metals, Semiconductors and Insulators

e′

Figure 10.9 Idealised representation of an F centre, an anion vacancy plus a trapped electron, in an alkali metal halide crystal electron-excess F centres, each consisting of an anion vacancy plus a trapped electron. It is believed that the colour centres in Blue John were formed when the ﬂuorite crystals were fortuitously located near to uranium compounds in the rock strata. Radioactive decay of the uranium produced the energetic radiation necessary to form colour centres. One of the best understood hole-excess centres gives rise to the colour in smoky quartz and some forms of 3þ amethyst. These minerals are essentially crystals of silica (SiO2) which contain small amounts of either Al or Fe3þ as substitutional impurities, Al0 or Fe0 . Charge neutrality is preserved by way of incorporated þ Si Si hydrogen as H . The colour centre giving rise to the smoky colour in quartz is formed when an electron is 5 þ liberated from an [AlO4] group by ionising radiation and is trapped on one of the H ions present. The reaction can be written as:

5 þ 4 ½AlO4 þ H !½AlO4 þ H

4 5 The colour centre is the [AlO4] group, which can be thought of as [AlO4] together with a trapped hole. The colour arises when the trapped hole absorbs radiation. The situation in amethyst, containing Fe3þ impurities, is similar. These crystals are a pale yellow colour due to the crystal-field splitting of the d-electron levels on the Fe3þ ions. In this form, natural crystals are known as 4 þ citrine, a semiprecious gemstone. On irradiation, [FeO4] groups form by interaction with H ions, as 4 5 described for [AlO4] above. The colour centre, an [FeO4] group containing a trapped hole, is able to absorb light, giving the crystals the purple amethyst coloration (Figure 10.10). If these crystals are heated to high temperatures the purple coloration is lost is and replaced by pale yellow crystal-field colours due to Fe3þ . This technique is sometimes used to convert relatively inexpensive amethyst into an artificial form of the rarer and more costly semiprecious stone citrine. There is a great deal of interest in the formation of colour centres in minerals by irradiation. In part this is because of the possibility of creating an impressive gemstone from an inexpensive precursor. The most widely Colour and the Optical Properties of Materials 432

Figure 10.10 Crystals of amethyst from Brazil. The purple coloration is due to hole centres, the intensity of the purple hue being proportional to the number of centres present in a crystal

available irradiated stone is topaz, Al2SiO4(F,OH)2. Normally, good-quality topaz is clear and of little value. 7 4 The structure contains [AlO4F2] octahedra which, like the [AlO4] above, are able to form stable colour centres under irradiation. These endow the stones with a beautiful blue colour (Figure 10.11). Although the exact cause of the coloration is not completely clariﬁed, the rutile form of the white pigment titanium dioxide (TiO2) seems to be coloured by hole centres formed as a consequence of the incorporation

Figure 10.11 Blue topaz stones. The blue colour is induced in the colourless topaz crystals by irradiation 433 Colour in Metals, Semiconductors and Insulators

Figure 10.12 Crystals of rutile, one form of titanium dioxide (TiO2), coloured yellow–orange by the inclusion of small quantities of gallium trioxide (Ga2O3)

3þ of colourless Ga ions. When crystals of rutile are heated with gallium oxide (Ga2O3), small quantities of Ga3þ impurity are readily incorporated into the structure and initially clear single crystals of rutile take on a yellow orange colour (Figure 10.12). The impurity Ga3þ ions enter the rutile structure and substitute for Ti4þ 0 ions in octahedral sites to form GaTi defects. These impurities have an effective negative charge, allowing them to trap positively charged holes. The liberation of the holes absorbs energy towards the violet end of the spectrum and colours the crystals yellow orange. Colour centres can give rise to a variety of useful colour effects. The oxide SrAl2O4 is a long-life phosphor giving a green output colour when doped with B, Eu2þ and Dy3þ . The origin of the colour lies in two complex colour centres formed by the impurity cations. The structure of this phase is a distorted form of tridymite, which 2þ 3þ is composed of corner-linked AlO4 tetrahedra that enclose Sr ions in the cavities so formed. The B 3þ 3þ 2þ substitutes for Al to create BO4 tetrahedra and BO3 triangular groups. The Dy substitutes for Sr to . 2þ 20 form DySr defects. Charge is balanced by the creation of Sr vacancies, VSr:

ð Þ! . þ 20 þ þ Dy2O3 3SrAl2O4 2DySr VSr 6AlAl 12OO

ð 0 .Þ ð 20 Þ Two complex centres form: Dy-- BO4-- VSr-- h , which are hole centres formed thermally from Dy-- BO4-- VSr , ð . 0Þ ð 2 .Þ and BO3-- VO -- e , which are electron centres formed from BO3-- VO under violet light. Under normal conditions, the electron and hole centres are metastable and the holes and electrons gradually recombine. The þ energy liberated is transferred to the Eu2 ions, to give a green ﬂuorescence. As there is no radioactivity Colour and the Optical Properties of Materials 434

Mg2+

O2-

2• ′ VO + e

OH-

þ Figure 10.13 An Fs (OH ) centre on an MgO (100) surface (schematic) involved, these materials can be used for luminous dials on clocks and watches, replacing a historic use involving radioactive materials, or as cold-light displays.

10.5.3 Surface colour centres

The concept of colour centres has been extended to surfaces to explain a number of puzzling aspects of surface 2 . reactivity. For example, in oxides such as MgO an anion vacancy carries two effective charges, VO . These vacancies can trap two electrons to form an F centre or one electron to form an Fþ centre. When the vacancy is þ located at a surface, the centres are given a subscript s, i.e. Fs represents a single electron trapped at an anion vacancy on an MgO surface. As the trapping energy for the electrons in such centres is weak, they are available to enhance surface reactions. þ The concentration of Fs centres can be increased by irradiation with energetic radiation such as X-rays or ultraviolet light, as well as by reaction with hydrogen. This latter reaction has led to the suggestion that several þ new colour centres could form involving hydroxyl. The Fs (OH ) centre is imagined to form in the following way. A hydrogen atom reacts on the surface to form a hydroxyl group, OH . This leaves the surface to link to a nearby metal cation in the exposed surface, at the same time creating an oxygen vacancy and leaving a trapped þ electron to create an Fs (OH ) defect (Figure 10.13). The properties of defects of this type are difﬁcult to determine experimentally, although absorption spectra do give information about electron or hole binding energies. Much information is obtained by calculation, using density functional or other quantum computational methods. In this way, the relative stabilities of defects on plane faces, steps, terraces and corners is being explored.

10.5.4 Complex colour centres: laser action

The fabrication of lasers based upon colour centres adds a further dimension to the laser wavelengths available. Ordinary F centres do not exhibit laser action, but F centres that have a dopant cation next to the anion vacancy are satisfactory. These are typiﬁed by FLi centres, which consist of an F centre with a lithium ion neighbour (Figure 10.14a). Crystals of KCl or RbCl doped with LiCl, containing FLi centres, have been found to be good laser materials yielding emission lines with wavelengths between 2.45 and 3.45 mm. A unique property of these crystals is that in the excited state an anion adjacent to the FLi centre moves into an interstitial position (Figure 10.14b). This is type II laser behaviour, and the active centres are called FLi (II) centres. These complex defects are introduced in a series of steps. Take KCl doped with Li as an example. Initially, KCl crystals are grown from a solution containing LiCl as an impurity. The Liþ cations form substitutional 435 Colour in Metals, Semiconductors and Insulators

e′ Li

(a)

e′

(b)

Figure 10.14 Schematic diagram of FLi colour centres in KCl: (a) ground-state FLi centre; (b) excited-state type II FLi centre responsible for laser output

LiK impurity defects distributed at random throughout the crystal. F centres are introduced by irradiation using þ X rays. These are not usually located nextto a dopant Li cation. Toconvertthe F centres into FLi (II) centres the crystal is subjected to a process called aggregation. In this step, the crystals are cooled to about 10 C and then exposed to white light. This releases the electrons trapped at the F centres, leaving ordinary anion vacancies, which are then able to diffuse through the crystal before recombining with the electrons once more to reform the F centre. Ultimately, each vacancy ends up next to an Liþ ion. At this position it is strongly trapped and further diffusion is not possible. Recombination with an electron forms the FLi centre required. This process of aggregation is permanent if the crystal is kept at 10 C and in this state the crystal is laser active.

10.5.5 Photostimulable phosphors

Photostimulable phosphors are widely used in X-ray imaging, particularly by dentists, where it has largely replaced X-ray film recording. In dental X-ray imaging, a plate covered with a thin layer of a phosphor is placed into the mouth and exposed to X rays. The X rays generate electrons and holes that are trapped at defects in the phosphor and do not recombine. This process is said to generate a latent image in the phosphor. Subsequent irradiation of the plate with a light source of the correct wavelength gives the electrons or holes sufficient Colour and the Optical Properties of Materials 436 energy to escape the trapping defects, allowing them to recombine. This leads to light emission, usually via energy transfer to an activator. The subsequent fluorescence is recorded as a digital image. The number of trapped electron and holes and, therefore, the amount of fluorescent emission is proportional to the X-ray intensity. The optical image thus records accuratelythedegreetowhichtheXrayshavepenetrated the subject. A suitable phosphor must be very efficient at absorbing X rays, to lower any danger to the patient, and must have a high luminous output when irradiated after the X-ray exposure. There should be no afterglow, which would seriously degrade image resolution. In addition, the phosphor must be reusable. The first commercial material to fulfil these requirements, introduced in 1983, was BaFBr doped with Eu2þ . Since that time many other systems have been explored for use in X-ray imaging, especially other binary and ternary alkali halides doped with Eu2þ as activator. At present (2010) the detailed mechanism by which these phosphors work is not altogether clear. However, it is well established that an important component of the process is the formation of F centres. These are produced as a result of X-ray irradiation and are similar to those in alkali halides described above (Section 10.5.1), consisting of an anion vacancy together with a trapped electron. These make þ up the electron trapping centres. For the commercial phosphor BaFBr:Eu2 , the radiation used to liberate the electrons trapped at the F centres is usually from a helium neon laser at 633 nm. The electrons, promoted to the conduction band, can then recombine with holes in the valence band. The energy is transferred to Eu2þ ions which give out visible light at 420 nm (see Section 9.4). This is not the only proposed mechanism of light emission. In some phosphors it has been suggested that X-ray irradiation forms Eu3þ ions, which are equivalent to Eu2þ together with a trapped hole. Electrons liberated by irradiation then recombine with holes at an Eu3þ ion without involving energy transfer. The result is blue emission from Eu2þ as before. Considerable research is ongoing to unravel the mechanisms by which photostimulable phosphors produce light and to produce new phosphors with greater resolution.

10.6 The Colours of Inorganic Semiconductors

10.6.1 Coloured semiconductors

In an (inorganic) insulator, the upper conduction energy band is completely empty and the lower energy valence band is completely filled. As the band gap shrinks, a profound change comes over the colour (and electronic properties) of the insulator, which gradually becomes an (inorganic) semiconductor. Intrinsic semiconductors5 have a similar band picture to insulators except that the separation of the empty and filled energy bands is small. How small is small? The original definition of a semiconductor as a poor electrical conductor suggests that the band gap must be such that some electrons have enough energy to be transferred from the top of thevalence band to the bottom of the conduction band at room temperature. The band gap of silicon, one of the most important intrinsic semiconductors, is approximately 1.1 eV, and this may be taken as representative for semiconductor band gaps. A remarkable property ofintrinsic semiconductors is that each electron transferred will leave behind a hole in the valence band. In an intrinsic semiconductor, both holes and electrons contribute equally to the electrical conductivity. In the idealised band picture, both of these particles are able to move through the solid in unhindered fashion, and so are often called free electrons or free holes.

5 Intrinsic semiconductors are pure materials with, ideally, no impurities. The majority of semiconductors used in devices are extrinsic semiconductors, in which impurities (dopants) are deliberately added to confer speciﬁc electronic properties on the material. p type semiconductors are doped so as to electrically conduct mainly by way of holes. n type semiconductors are doped so as to electrically conduct mainly by way of electrons. 437 Colour in Metals, Semiconductors and Insulators

The colour of a pure semiconductor is, to a first approximation, governed by the band separation energy. When the energy gap is relatively large, light photons are not energetic enough to excite an electron from the valence band to the conduction band and so are not absorbed. The material will appear transparent to the eye. (This is so for diamond, with a band gap of approximately 5.5 eV,and titanium oxynitride (TiON), with a band gap of approximately 4.12 eV, although these compounds are generally regarded as insulators rather than semiconductors.) On the other hand, if the energy gap is quite small, corresponding to the infrared region, the semiconductor will absorb the entire visible spectrum and take on a black or metallic appearance. Silicon, with a band gap of approximately 1.1 eV,is typical of this group. In powder form it is black. Single crystals appear to look metallic. If the band gap energy falls in the visible, between approximately 1.77 and 3.10 eV, the semiconductor will absorb all photons with energy greater than the band gap energy and not those with a smaller energy. This will cause the material to be strongly coloured. For example, the pigment vermilion, which is produced from the mineral cinnabar, mercuric sulfide (HgS), has a band gap of approximately 2.0 eV.This energy corresponds to the red orange region of the spectrum. All shorter wavelengths than this are associated with more energetic photons, and these will be absorbed. These are the yellows, greens and blues. The colour perceived will be due to the photons with energy less than 2.0 eV, which are not absorbed. These are the reds and oranges (Figure 10.15a). The pigment cadmium yellow, cadmium sulfide (CdS), has a band gap of 2.42 eV, which corresponds to the green blue part of the visible. Photons of lower energy, red, orange, yellow and green, will not be absorbed, while the higher energy blue, violet and indigo will be. The net result is that the pigment appears yellow to the eye. Almost all coloured sulfides have figured as artist’s pigments in one context or another in earlier centuries. For example, a less widely used pigment these days is orpiment, arsenic trisulfide (As2S3). The mineral name is a corruption of the Latin auri pigmentum, golden paint, and it is also known as the artist’s colour King’s yellow. It is readily prepared as a canary yellow precipitate when hydrogen sulfide gas is passed into solutions containing As3þ ions. The pigment has fallen into disfavour because of its toxicity and tendency to give off poisonous vapour when exposed to damp air. Apart from sulfides, many other materials are brightly coloured in the same way. These include the decorative hard coating materials titanium nitride (TiN), which is golden (often seen as gold-coloured hard tips on drill bits), zirconium nitride (ZrN), which is yellow green, tantalum nitride (TaN), which is blue grey, and titanium carbide (TiC) and tungsten carbide (WC), both of which are dark grey. These materials, as well as some metal sulfides show a similarity to metals, both visually and electronically. This can happen if a large number of electrons are present in the conduction band. In this case the electrons may take on properties similar to those of metals (Section 10.15). The most widely known example of this similarity is found in the compound pyrite, FeS2, also known as fool’s gold (Figure 10.15b). The physical properties are not at all metallic, however; pyrite is brittle rather than malleable, like gold is. Conductivity is still by way of both electrons and holes, whereas in a metal only electrons are important. Admixture of copper sulfide (CuS) with pyrite produces the mineral chalcopyrite, with a nominal formula Cu2Fe2S4. This material also has a metallic appearance and takes on a variety of golden or purplish hues, depending upon the exact composition, for the same reason (Figure 10.15c). As with the insulators described above, the band gap of semiconductors tends to decrease with temperature, leading to thermochromism.

10.6.2 Transparent conducting oxides

The electrical conductivity of a semiconductor depends upon the number of holes and electrons present. Doping is widely used to modify these populations and so alter the measurable conductivity. If this can be achieved in a semiconductor with a fairly large band gap the conductivity may be appreciablewhile the material Colour and the Optical Properties of Materials 438

Figure 10.15 The colours of semiconductors: (a) cinnabar, mercuric sulﬁde (HgS); (b) pyrite (FeS2), fool’s gold; (c) chalcopyrite, nominally Cu2Fe2S4 439 Colour in Metals, Semiconductors and Insulators

Figure 10.15 (Continued)

remains transparent. This is the situation in transparent conducting oxides (TCOs), sometimes referred to as transparent metals, which are widely used as transparent conducting electrodes. The best known of these materials is indium oxide (In2O3) doped with between 5 and 15 mol% tin oxide (SnO2), known as indium tin oxide or ITO. Surprisingly, in view of the importance of this material, there are (2010) ongoing attempts to explain its electronic properties. (Indeed, there is still disagreement about the true band gap of pure In2O3, which is reported to vary from about 2.8 to 3.75 eV.) Irrespective of the true value, In2O3 is a lemon yellow colour and has an absorption spectrum very similar to that of WO3 (Figure 10.1c). The absorption spectrum just creeps into the visible at the blue end of the spectrum, giving a resultant yellow tone to the bulk solid. The transparent electrode material ITO is also a pale yellow colour in bulk, but when prepared as a thin ﬁlm it appears transparent to the eye. Incorporation of SnO2 into In2O3 leads to the formation of defects which in turn leads to the increase in conductivity whilst retaining the large band gap that makes the oxide transparent. Whilst there is still considerable uncertainty about the nature of these defects and how the doping inﬂuences the band structure of the host In2O3, the following broad-brush picture describes the state of affairs that is believed to occur. Colour and the Optical Properties of Materials 440

4þ 3þ . 6 The Sn ions are considered to mainly occupy In sites forming SnIn point defects. The additional 2 20 oxygen is accommodated as interstitial O ions, Oi :

ð Þ! . þ 20 $ð Þx 2SnO2 In2O3 2SnIn Oi 2SnIn Oi

There is some evidence to suppose that the tin and oxygen interstitial defects may aggregate into a neutral x defect complex (2SnIn Oi) rather than remain isolated. In either case, the number of interstitial oxygen defects will vary with the ambient oxygen pressure during thin-ﬁlm preparation according to the reversible equation:

0 O2 $ 1 O þ 2e0 i 2 2

This means that interstitial oxygen defects are preferred at higher ambient oxygen pressures, while electrons are produced at lower pressures. This accounts for the fact that highly conducting oxide films are prepared under reducing conditions; that is, at relatively low oxygen partial pressures, incorporation of SnO2 into In2O3 leads to the production of electrons. These electrons enter the conduction band to enhance the n-type conductivity of the oxide. As the dopant concentration rises, the number of electrons in the oxide increases. At a dopant concentration of about 2 1019 cm 3 the electrons behave as free electrons, rather similar to those in a metal. Semiconductors that are so heavily doped that the conductivity approaches that of a metal are called degenerate semiconductors. The band gap, although varying with dopant concentration, remains wide enough for the material to appear transparent in thin-film form. A number of other n-type transparent oxide conductors have been found, including tin oxide (SnO2) doped with F, zinc oxide (ZnO) doped with Al2O3, and a number of oxides with structures related to that of fluorite (CaF2). Unfortunately, a matching transparent p-type oxide conductor has not yet been found, although delafossite- 3þ structure oxides CuM O2, including CuGaO2, CuInO2 and CuScO2, have potential in this respect. Such a material is considered to be important because it would allow for highly desirable transparent electrodes at each face of a light-emitting device (Sections 10.8 and 10.11).

10.7 The Colours of Semiconductor Alloys

Band gaps of semiconductors can be finely tuned by making solid solutions spanning the composition range between two isostructural parent phases. This can be illustrated with respect to cadmium sulfide (CdS) and the very similar cadmium selenide (CdSe). Both of these compounds adopt the wurtzite structure, one of the forms of zinc sulfide (ZnS). CdS, with a band gap of 2.42 eV, absorbs high-energy photons from violet to blue. CdSe has a smaller band gap of 1.74 eVand absorbs all the visible wavelengths. It appears black to the

6 The Kroger Vink point defect notation is used; see this chapter’s Further Reading, for details. 441 Colour in Metals, Semiconductors and Insulators eye. The sulfur and selenium atoms in these two compounds are of a similar size, which allows one to replace the other readily. If a solid solutionismadewithageneralformulaCdS1 xSex the band gap gradually changes from that appropriate to CdS at x ¼ 0 to that appropriate to CdSe at x ¼ 1.0. At x ¼ 0 the photons absorbed are only those in the violet to blue region of the visible, but, as x increases, the range of absorbed photons moves towards red and infrared. The colour perceived gradually changes from yellow at x ¼ 0 to orange to red and ultimately to black as x increases. The material CdS0.25Se0.75 is the pigment cadmium orange. Most isostructural pairs of semiconductors can form solid solutions in the same way. In these instances, the band gap can be manipulated at will. Note, though, that the dependence of band gap upon composition is not linear, but tends to follow a shallow curve. For example, the band gap of the important semiconductor system gallium nitride indium nitride (GaN InN) is given by:

Eg ðalloyÞ¼xEg ðGaNÞþð1xÞEg ðInNÞxð1xÞb

where Eg represents the relevant band gap and b is called the bowing coefﬁcient or bowing parameter. Inserting experimental values for the band gap of GaN (3.30 eV) and InN (0.61 eV) and a bowing coefﬁcient of 1.43 gives the quadratic function (Figure 10.16):

2 Eg ðalloyÞ¼0:61 þ 1:26x þ 1:43x

It is seen that this system spans the visible. In this context, the isostructural insulator aluminium nitride (AlN) has a band gap of 6.1 eV.Alloys can be fabricated with GaN that take the emission into the ultraviolet. The alloy range AlN GaN InN can, therefore, give an output anywhere between the deep ultraviolet and the infrared. Many of the semiconductors mentioned above also form alloys with varying colours. The titanium carbonitride TiCxNy, with (x þ y) 1, varies from gold to red. The closely related zirconium carbonitride ZrCxNy takes on hues between silver, gold and violet, depending upon composition. A change of band gap with temperature can lead to a change in the perceived colour of the phase, giving rise to thermochromism.

10.8 Light Emitting Diodes

10.8.1 Direct and indirect band gaps

Electrons in the conduction band can gain energy by dropping back to the valence band and recombining with a hole. This energy is frequently released as a photon, and so semiconductors can act as lamps given a continuous power input to maintain the supply of charged particles. Such materials display electroluminescence, which is light emission following an input of electrical energy. The colour of light emitted in this way is naturally inﬂuenced by the band structure of the material. However, the band structure must be considered in more detail than before to understand emission from semiconductors. The real band structure of a solid can be envisaged as a series of undulating surfaces, resembling stacked sheets, which deﬁne the allowed energy states accessible to electrons and holes, usually plotted as a graph of Colour and the Optical Properties of Materials 442

2 Band gap / eV

0 0 0.2 0.4 0.6 0.8 1.0

Fraction x in GaxIn1-xN

Figure 10.16 The band gap of the alloy series GaN–InN as a function of the composition

energy versus the wave vector k of the electron defined as 2p/l, where l is the electron wavelength.7 Each surface consists of a series of hills and valleys, not the flat bands used in the simple descriptions given above. If the highest ‘peak’ in thevalence band corresponds to the lowest ‘valley’ in the conduction band, an electron can absorb a photon and be promoted directly across the band gap, leaving a hole in the valence band. This characterizes a direct band gap material (Figure 10.17a). The reverse process is also possible: an electron in the conduction band can emit a photon and directly recombine with the hole in the valence band (Figure 10.17b). If the peak in the valence band does not correspond to the lowest point of the conduction band that is, if the two energy features are displaced with respect to each other an electron can only be promoted from the lower to the upper band if it is given an increment of momentum k. This ‘sideways kick’ is equivalent to the addition of a phonon (a quantum of lattice vibration) to the process. The electron must interact with both a photon and a phonon simultaneously to jump the band gap (Figure 10.17c). This situation characterizes an indirect band gap material. The reverse process, in which an electron gives up both a photon and a phonon so as to recombine with a hole in the valence band, is of low probability, and generally indirect band gap materials do not emit radiation efficiently (Figure 10.17d). Energy is lost instead by internal energy conversion, i.e. nonradiative transitions. The nature of the band gap, direct or indirect, is of vital importance when luminous efficiency is concerned. Indirect band gap materials are generally very poor light emitters.

7 As the momentum of the electron is equal to kh/2p, the k axis is often labelled as momentum. 443 Colour in Metals, Semiconductors and Insulators

conduction band

ν h direct Eg transition hν

valence band

-k 0 +k -k 0 +k (a) (b)

conduction band

phonon

indirect Eg transition photon

valence band

-k 0 +k -k 0 +k (c) (d) Figure 10.17 Direct and indirect band gap materials. (a), (b) A direct band gap material can absorb and emit photons equally efﬁciently. (c) An indirect band gap material requires that a photon and a phonon combine to promote an electron. (d) The reverse process is of low probability and indirect materials do not make satisfactory light emitters

10.8.2 Idealized diode structure

To make an electroluminescent light-emitting device it is necessary to pump electrons into the conduction band and holes into the valence band so that recombination can occur continuously. The arrangement suited to this is that of a semiconductor LED. These are formed by the juxtaposition of a region of n-type and p-type semiconductor, grown into a single crystal. In the p-type region of the material the semiconductor has been doped with acceptors so that the top of the valence band contains a high population of mobile holes. It is, in fact, a hole conductor. In the n-type region the semiconductor has been doped with donors, so that the material contains a population of mobile electrons at the bottom of the conduction band. This region is an electron conductor. When a p-type region abuts an n-type region, electrons move into the p-type region from the n- type side and holes move into the n-type region from the p-type region, by diffusion. Most of the displaced electrons and holes recombine and so are eliminated. However, as electrons leave the n-type region, positively charged donor atoms are left behind, while negatively charged acceptor atoms are left in the Colour and the Optical Properties of Materials 444 p-type region as holes leave. This net imbalance in the charges present is called the space charge.Theresult of the changes is to create an electric potential, the contact potential,orbuilt-in potential of about 0.3 V.At equilibrium, the energy bands have been shifted give a distorted band structure in the junction region (Figure 10.18a). The transition region, which is also called the depletion region or active region, has a width of about 1 mm in an ordinary diode. At equilibrium (thermal and electrical) there will still be an exchange of carriers at the junction, but the current in each direction will be the same. Dynamic equilibrium holds. This changes when a voltage is applied across the junction. An applied voltage, which will drop across the transition region, because of the absence of mobile charge carriers, can be applied with the positive side connected either to the p-type region or to the n-type region. The arrangement in which the positive voltage is connected to the p-type region is called forward bias. This causes the potential barrier to be reduced. Under a forward bias there is a rapid increase in the current ﬂowing across the junction. Electrons and holes now enter the junction

junction region

conduction conduction band band

photon

electrons electrons holes + holes + + + photon p-type + p-type + valence valence band band

(a) n-type (b) n-type +

current

p-type junction region n-type

photon current

(c) _ Figure 10.18 A semiconductor LED (schematic): (a) equilibrium situation; (b) under a forward bias, light is emitted in the junction region; (c) schematic device construction of a homojunction LED 445 Colour in Metals, Semiconductors and Insulators and recombine. The energy released, which is approximately equal to the band gap, can appear as light (Figure 10.18b). The number of electrons and holes in the device and entering the active region is continually replenished by the external power supply. The size of LEDs is remarkably small and often compared with a pencil point or a grain of salt, although dimensions much less than this can easily be realized. The simplest (conceptual) device conﬁguration is just a planar slab, made of the same material, one part doped to be n-type and one part to be p-type, to form a homojunction LED (Figure 10.18c).

10.8.3 High-brightness LEDs

Of course, this simplified account does not do justice to the immense amount of hard work needed to produce the efficient LEDs now available at little cost. The initial studies were made on gallium arsenide in the early 1960s. This semiconductor is a direct band gap material with a band gap of 1.42 eV, giving out infrared wavelengths. The first challenge was to convert this to visible. The material gallium phosphide (GaP) was a suitable contender, with a band gap of 2.26 eV, but this compound is unfortunately an indirect band gap material. However, combining GaAs with small amounts of GaP resulted in a direct band gap alloy, and compositions close to GaAs0.6P0.4 were used to obtain successful light emission in the red region of the spectrum. The first red LEDs were commercially available in 1962, although they were not very efficient and certainly did not have a good brightness. They made use of a single semiconductor doped either p- or n-type, and are homojunction devices. Unfortunately, the band gap of the GaAs GaP alloys changes from direct to indirect part way across the composition range. This means that not all of the composition range can be utilized, so that the only colours available were towards the yellow orange red end of the spectrum. The brightness of LEDs has been improved dramatically since then. This has been due to a number of major advances. First, crystal defects, in particular dislocations running through the active layer, have been greatly reduced. These defects provide sites at which electrons and holes can combine nonradiatively, hence lowering device brightness. Second, more complex alloy systems have been developed to give a broader spectral range. Thus, red, yellow and orange emitters rely on quaternary alloys of gallium arsenide (GaAs), aluminium phosphide (AlP), gallium phosphide (GaP) and indium phosphide (InP) with typical composition (AlxGa1 x)0.5In0.5P and a direct band gap between 1.9 and 2.26 eV, as a function of the Al content. The light- emitting region, has been confined between semiconducting slabs with different overall composition and of lower refractive index in heterojunction devices, which channel the light more effectively using total internal reflection (Figure 10.19a). The active layer has also been fabricated into single or multiple quantum-well configurations (see Section 10.9) approximately 2 mm thick. This confines the holes and electrons to a narrow spatial region, increasing the likelihood of recombination and again increasing brightness. Reflecting layers, either multiple thin-film mirrors or simple reflecting cups in which the chip is mounted, also improve apparent brightness. In 1993 the reds and yellows of the GaAs GaP system were supplemented with alloys in the GaN InN system, which are direct band gap alloys across the whole of the composition range and are able to provide blue and green light. (Note that, by including direct band gap AlN in the system, alloys can be fabricated that are able to give an output anywhere between the deep ultraviolet and the infrared.) Mechanical design changes are also important. In a planar material, light can only escape if it meets a face at less than the critical angle given by Snel’s law (Section 2.2). In 1998, the design of the light-emitting chip was changed to a truncated pyramid, with sides at 35 to the vertical, so as to optimize the escape of photons (Figure 10.19b). Colour and the Optical Properties of Materials 446

electrode

μ n-Al0.8Ga0.2As layer, ~30 m μ active layer, p-Al0.35Ga0 65As, ~2 m

μ p-Al0.75Ga0 25As layer, ~100 m

(a) reflecting substrate

electrode

n-GaP ~ 200 μm 35° active layer, AlGaInP ~2 μm p-GaP ~ 55 μm (b)

Figure 10.19 LED structure: (a) typical planar heterojunction LED; (b) truncated prism design LED, cut at an angle of 35 to optimize brightness

10.8.4 Impurity doping in LEDs

The colours produced in diamond by impurities suggest that it might be possible to place impurities into the band gap of LEDs and manipulate the system so that new colours are emitted. It has been found that this can be done successfully with lanthanoid elements introduced into silicon, such as silicon doped with erbium (Si:Er). However, the most successful colour-producing devices that have been fabricated contain lanthanoids introduced into gallium nitride (GaN). The advantage of using lanthanoids is that the important 4f energy levels of the atoms are shielded from the influence of the surrounding host structure by filled 5s2 and 5p6 orbitals. The energy levels remain narrow and the colours produced only spread over a very narrow range of wavelengths. Although light emission from these dopants can be obtained via photoluminescence (Chapter 9), for LED use it is more desirable to populate the upper levels by applying an electrical potential to the semiconductor. Electrons can then continuously fill the upper energy levels of the lanthanoid dopant and light will be emitted as these excited atoms return to the ground state. Electroluminescent devices GaN:Pr and GaN:Eu yield red output, GaN:Er produces green and GaN:Tm violet (Figure 10.20). It is clear that a combination of these three devices (sometimes referred to as light-emitting devices or, confusingly, LEDs) could be employed for flat- screen displays. The energy levels populated can be controlled by variation of the electrical input to the gallium nitride diode. Thus, it has been demonstrated that GaN:Er can also emit in the infrared at 1538 nm. This is a very convenient output, because it matches both the minimum attenuation of silica optical fibres (Section 2.9) and an important energy region of Er-doped optical-fibre amplifiers (Section 7.17).

10.8.5 LED displays and white light generation

LEDs generate light of speciﬁc colours. Impressive full-colour displays using LEDs have been built using millions of small red, green and blue units. From this point of view, using three LED colours to give the 447 Colour in Metals, Semiconductors and Insulators

3.5 GaN band edge

3.0

1 G4 2.5 3P 0 2 5 H11/2 D1 4 5D S3/2 1D 0 2.0 2 477 nm 647 nm

Energy / eV 543 nm 3 650 nm 537 nm H4 1.5 558 nm 600 nm 1 4I G4 11/2 3H 1.0 5 621 nm 3 4 F3 I13/2 3 3F H4 0.5 2 663 nm 3H 6 7 1538 nm 3 F3 H5 7 F2 3 4I 3H 0 H4 7F 15/2 6 Pr3+ Eu3+ 1 Er3+ Tm3+ þ þ þ þ Figure 10.20 Energy levels of some lanthanoids in GaN: (a) Pr3 , (b) Eu3 ; (c) Er3 , (d) Tm3

impression of white light, by virtue of additive colour mixing (Section 1.10) does not give a particularly good colour rendition, and four colours, red, yellow, green and blue are more satisfactory. The displays have no size limitations and the intensity produced by each LED is also sufficient to make them easily visible in daylight. There are, though, a number of limitations. The efficiency of the LEDs varies; in particular, green emitters are less efficient than red or blue. This means that the numbers of LEDs selected for the display must take efficiency into account. Moreover, efficiencies vary with the age of the LED, so this again must be corrected if the display is to retain high brightness and good colour rendition over time. For many purposes white light is essential. The simplest way to make a white light is to combine appropriate LEDs, which, when viewed from a distance, appear white by way of additive coloration. However, while this is satisfactory for displays, it is not satisfactory for most lighting purposes. For this, the commonest way to make white light is to coat a monochromatic LED with a phosphor. It is these devices that are normally termed ‘white LEDs’. The commonest white LED configuration is to use a high-intensity (Ga,In)N blue-emitting LED coated with a yellow-emitting phosphor. At present, this is most often yttrium aluminium garnet (Y3Al5O12) doped with Ce3þ , which gives a rather broad yellow emission. The coating is sufficiently thin to allow a certain amount of blue light to be transmitted. This, together with the yellow luminescence, creates a cool bluish white colour. These white LEDs are found in many applications, including cycle lamps, flash lamps, traffic lights, headlamps, tunnel illumination and so on. The rather cold light is not entirely suitable for indoor lighting, which ideally needs to be warmer. For this, two phosphors are used, a red and a green emitter in tandem with 2þ a (Ga,In)N blue-emitting LED. The orange red phosphor is the nitrosilicide Sr2Si5N8 doped with Eu . The red orange tone can be adjusted by replacing some of the Sr by Ca. The green phosphor used is often Colour and the Optical Properties of Materials 448

2þ SrSi2O2N2 doped with Eu . Once again, colour tuning is possible by substitution of the Sr by Ca. Blue light is contributed by the LED itself. There is considerable work in this area, and new phosphors are continually being tested. Moreover, phosphors for use with ultraviolet-generating diodes are also available. The improvements in white LED lamps are certain to be considerable in the near future.

10.9 Semiconductor Diode Lasers

If a forward bias voltage is applied to a suitable p n junction it will function as an LED. In principle, it is very easy to turn an LED into a semiconductor laser, a diode laser. This is because (at least in principle), a population inversion is achieved by using a heavily doped n-type region with respect to the p-type region. The population inversion is generated in the following way. When a low forward current is applied to the diode, a small and roughly equal number of holes and electrons enter the junction region, recombine and emit light. This is normal LED behaviour and produces a radiance more or less in proportion to the magnitude of the current. However, at some critical threshold voltage, far more electrons start to enter the junction region than holes. This is because the hole transport into the junction region is at a plateau, the height of which is controlled by the relatively low doping level. However, the electron transport into the junction region can continue to rise because the n-type segment is heavily doped and has a far higher plateau. At the point when the electron numbers outweigh the number of holes, a population inversion occurs and the LED becomes a laser (Figure 10.21a). The light is emitted from the active region as in an LED. Stimulated emission is achieved by using carefully polished crystals so that any photon emitted will be reflected to and fro in the junction to promote the laser avalanche. The change from LED to laser operation is marked by a large increase in both output and efficiency as the current passes a point termed the threshold current (Figure 10.21b). Semiconductor lasers are small, being of comparable size to ordinary LEDs. Diode lasers were developed in tandem with LEDs, and it is not possible to disentangle the evolution of one device from the other. Initial diode lasers were made from materials related to gallium arsenide (GaAs) or indium phosphide (InP). The output wavelengths fall into the ranges of approximately 630 980 nm for GaAs-derived systems and 1300 1550 nm for InP-derived systems. An early problem was the construction of a resonant cavity to ensure laser output rather than LED output. This was solved by polishing the ends of the semiconductor crystals making up the LED/laser. These first homojunction devices were not very efficient and operated best at liquid-nitrogen temperatures. Since then a large number of device structures have been explored, including the first device architecture that led to successful room-temperature operation, in which a double heterojunction construction was employed. This consists of a series of layers of different materials, such as n- and p-GaAlAs alloys of wide band gap surrounding an active layer of narrow band gap alloy. The buried heterostructure design, in which the active region is confined to a narrow strip, has been successfully used for many common laser devices. Photons emitted in the active layer are confined to this region due to refractive index differences between the surrounding alloys. This confinement enhances stimulated emission and gives a more collimated beam. Semiconductor laser diodes are ubiquitous. Perhaps the widest distribution is in barcode readers, found in every store. In the home, lasers are used for the recording and playing of CDs (785 nm GaAlAs red lasers), DVDs (650 nm GaAlAs lasers), and Blu-ray and HD-DVD discs (405 nm InGaN lasers) (Section 3.1). Laser measuring equipment is available for ordinary tasks such as room dimension measurement. Laser pointers are commonplace (670 nm GaAlAs or 650 nm AlGaInP) and are a good source of laser light for home experiments (e.g. see Section 6.7). Fibre-optic communications (Section 2.9) use semiconductor laser light to carry information. The list could be continued! 449 Colour in Metals, Semiconductors and Insulators

current

p-type polished face polished face

photons photons

n-type

current

(a) _

laser region Light output

LED region

threshold Current (b) current Figure 10.21 Semiconductor diode lasers. (a) Schematic device construction of a homojunction laser. (b) A current above a threshold value (ideally) converts an LED into a laser

10.10 Semiconductor Nanostructures

10.10.1 Nanostructures

Nanostructures are structures of a dimension that endows a solid with properties that are noticeably different from those of bulk material. The dimension at which this transformation becomes apparent depends upon the phenomenon investigated. In the case of thermal effects, the boundary occurs at approximately the value of 21 thermal energy kBT, which is about 4 10 J. In the case of optical effects, nonclassical (i.e. diffraction) behaviour is noted when the scale of the object illuminated is of the same size as a light wave, say about 5 10 7 m. For particles such as electrons, the scale is determined by the Heisenberg uncertainty principle, about 3 10 9 m. This is illustrated by the band gap of ZnO crystals, which starts to change from that of bulk material as the particle size approaches about 6 nm (Figure 10.2). In this section, the optical consequences of semiconductor nanostructures will be outlined. In these structures it is the electrons and holes that are under consideration and the length scales of importance are accordingly of the order of nanometres. The overall consequences of limiting the dimensions of a material can be understood in terms of outer electron interactions. The electrons on isolated atoms are associated with sharp energy levels located at the atom in question. The outer (valence) electrons on atoms in a molecule are delocalised over the molecule in Colour and the Optical Properties of Materials 450 molecular orbitals, but the energy of these remains mostly sharp. The outer electrons associated with an atom in a solid are spread over all of the atoms in the material and are associated with a transformation of narrow energy levels into energy bands. As noted earlier, in the reverse situation, as a solid is imagined to fragment into smaller and smaller units, the energy levels must change from typically bulk-like bands to more molecular and then atom-like sharp levels. Because of this, the manner in which the dimensions of a solid are constrained will have a major effect upon the resultant properties of the body. A thin layer of a material will have bulk properties modified towards atom- like properties in a direction normal to the layers. A thin layer of a semiconductor sandwiched between layers of a different semiconductor, called a quantum well, will show this behaviour (Figure 10.22a). Semiconductor electronic devices can increase the effect by stacking up several alternating thin layers to form multiple quantum well (MQW) structures (Figure 10.22b). Carbon nanotubes and nanorods of other compounds that are small on an atomic scale in two directions are known as quantum wires or nanowires (Figure 10.22c). A cluster of atoms, called a quantum dot, has properties approaching that of the isolated atoms. Electrons are distributed between energy levels that resemble atomic or molecular orbitals (Figure 10.22d). These structures have unique electronic and optical properties because of the way in which electrons are localised, or confined. An electron or hole bonding energy much greater than thermal energy characterises strongly confined charge carriers.

~10 nm

(a) (b)

Figure 10.22 Semiconductor nanostructures: (a) a single quantum well (SQW); (b) a series of quantum wells – a multiple quantum well (MQW) structure; (c) a quantum wire; (d) a quantum dot 451 Colour in Metals, Semiconductors and Insulators

10.10.2 Quantum wells

A single quantum well (SQW) is constructed by laying down a thin layer of a semiconductor with a smaller band gap within a semiconductor with a larger band gap. In this structure, the electrons and holes are essentially confined to the two-dimensional plane of the thin layers by the difference in the band structures of the two materials. The electrons are trapped at the bottom of the conduction band and the holes at the top of the valence band in the well. In an MQW structure, the electrons and holes are similarly trapped in the low band gap layers. For example, quantum well structures formed from a layer of gallium arsenide (GaAs) sandwiched in gallium aluminium arsenide (GaAlAs) trap electrons in the conduction band ‘valleys’ and trap holes in the valence band ‘hills’ located in the GaAs layers (Eg 1.42 eV) between the GaAlAs layers, with a band gap of approximately 1.75 eV, between 1.42 (GaAs) and 2.16 eV (AlAs) (Figure 10.23). The energy of an electron in a quantum well will be more or less the same as the energy in the bulk for the two directions in the plane of the confining sheet. In a direction normal to the sheet, the narrow dimension of the layer, a very rough estimate of the allowed energy levels available to an electron can be calculated by assuming that it is free and trapped by an infinite boundary potential. The electron, regarded as a wave, can only fit into the volume if the wave has a node at each boundary (Figure 10.24). In this case, the energy E of a free electron in a rectangular parallelepiped with edges a, b and c is given by

! 2 2 n2 2 ð ; ; Þ¼ h nx þ y þ nz ð : Þ E nx ny nz 2 2 2 10 1 8me a b c

where h is Planck’s constant, me is the mass of the electron and nx, ny and nz are the quantum numbers along the three axes. Exactly the same equation will apply to a free electron confined to a slab of material, although it * is better to replace the electron mass with the effective mass me . In the case of a quantum well, the electron is confined in one dimension, say x, and unconfined in two directions, which can be taken as y and z,soitis convenient to rewrite Equation 10.1 as

"# ! 2 2 2 n2 2 ð ; ; Þ¼ h nx þ h y þ nz ð : Þ E nx ny nz * 2 * 2 2 10 2 8me a 8me b c

The values of b and c can be taken as about 1 cm, while the value of a is about 10 8 m. The energy, therefore, is dominated by the ﬁrst term in Equation 10.2. This introduces a new set of energy levels, associated with electron waves trapped in the well (Figure 10.25a). The electron energy level in the 2 * 2 lowest, n ¼ 1, state is raised by h =ð8me a Þ compared with the base of the well. These energy levels are called electron subbands, and when the energy levels trap the electron strongly the electrons are said to be strongly conﬁned. * * Exactly the same equations apply to holes, when the effective mass mh replaces me . The energy levels that arise from trapped holes are called hole subbands. In a quantum well, the electrons and holes occupy these energy levels. The electrons in the upper energy levels can drop to the lower hole levels and emit photons (Figure 10.25b). The energy separation of these levels is greater than that of the bulk conduction band valence band energy gap Eg; hence, the photons will be of higher energy, or shorter wavelength, than the bulk. The emission is said to be blue shifted compared with the bulk, and the transitions are called interband transitions. Colour and the Optical Properties of Materials 452

GaAlAs GaAs GaAlAs

(a)

electrons

Eg GaAlAs Eg GaAs

holes

(b)VB

GaAlAs GaAs GaAlAsGaAs GaAlAsGaAs GaAlAsGaAs GaAlAs

(c)

Eg GaAlAs Eg GaAs

VB (d) GaAlAsGaAs GaAlAsGaAs GaAlAsGaAs GaAlAs GaAs GaAlAs Figure 10.23 Quantum wells (schematic). (a) A single quantum well (SQW) of gallium arsenide (GaAs) in a gallium aluminium arsenide (GaAlAs) alloy. (b) Schematic energy band sequence. (c) A multiple quantum well (MQW) structure formed from the same materials. (d) Schematic energy band structure of (c). CB, conduction band; VB, valence band; Eg band gap

The photon energy derived from an interband transition is:

EðphotonÞ¼hn ¼ Eg þ Eelectron þ Ehole 2 2 2 ð10:3Þ ¼ þ h n þ n Eg 2 * * 8a me mh 453 Colour in Metals, Semiconductors and Insulators

n = 3

n = 2 Energy

n = 1

-a/2 0 a/2 Figure 10.24 The ﬁrst three energy levels of an electron (or hole) trapped in a quantum well correspond to the three longest wavelength waves with nodes at the well boundaries

where h is Planck’s constant, n the frequency of the radiation emitted, Eg is the band gap of the bulk well * * material, a is the dimension of the quantum well, me is the electron effective mass and mh the hole effective mass. In the approximation that the effective mass of the electron and the hole are identical and equal to m:

n2h2 EðphotonÞ¼hn ¼ E þ ð10:4Þ g 4a2m

CB electron energy levels

hole energy levels VB (a) SQW

CB n = 1

n =1 n =1 n = 2 n = 2

n = 1 (b) VB Figure 10.25 Energy levels in a single quantum well (schematic): (a) electron (upper) and hole (lower) subbands; (b) interband transitions Colour and the Optical Properties of Materials 454

p-electrode

0.5 μm p-GaN:Mg

100 nm p-Al0.2Ga0.8N:Mg barrier layer

3 nm undoped In0.45Ga0 55N SQW active layer

4 μm n-GaN:Si

n-electrode Figure 10.26 Schematic diagram of the construction of a green-emitting LED containing a single quantum well (SQW) active layer

The selection rule for the transition is Dn ¼ 0; that is, transitions can only take place between levels with the same quantum number. (As with all selection rules, these are never perfectly obeyed, and transitions between levels with differing n values do occur infrequently, giving rise to weak lines in the emission spectrum.) Electrons can also be excited from one electron level, say n ¼ 1, to another electron level, say n ¼ 2, both levels lying in the electron subband. Holes can make similar transitions between levels in the hole subband. These transitions, which give rise to extra peaks in the emission spectrum, are known as intersubband transitions. Because the dimensions of the quantum well can be varied, the emission spectrum can be varied or tuned. This feature, in both quantum wells and in quantum wires and dots, discussed below, is called bandgap engineering. Quantum well structures are widely used in LEDs and laser diodes to improve device performance. They do this in a number of ways: by conﬁning electrons and holes into a limited space, so that recombination is more likely, and by guiding the photons emitted by virtue of the differing refractive indices of the materials. Typical of these device structures is the SQW structure used in the ﬁrst green-emitting LEDs (Figure 10.26). A change in the composition of the SQW active layer allows the colour emission to vary between 450 nm (blue) and 600 nm (yellow).

10.10.3 Quantum wires and quantum dots

The above considerations can be applied equally well to confinement in two or three dimensions, to give quantum wires and quantum dots. Fora quantum wirewith restricted dimensions along a and b, the free electron confined in an infinite potential well will have energy levels given by: ! 2 2 2 n2 2 2 ð ; ; Þ¼ h nx þ h y þ h nz ð : Þ E nx ny nz * 2 * 2 * 2 10 5 8me a 8me b 8me c

* where a and b are small and c is large. An analogous equation for holes, with effective mass mh can also be written. The case of a quantum dot, Equation 10.5, is retained, but the third dimension, c, is also small. For a roughly spherical quantum dot of radius r, Equation 10.3 then becomes: 2 2 2 ð Þ¼ n ¼ þ h n þ n ð : Þ E photon h Eg 2 * * 10 6 8r me mh 455 Colour in Metals, Semiconductors and Insulators

This indicates that the energy change is proportional to 1/r2. (Note, however, that these equations give only a rough estimate of the energy levels of the confined particles. The rigorous calculation of the energy levels available to electrons and holes in quantum dots can be assessed using quantum mechanical routines that take into account not only the wave functions of the constituent atoms, but also the strain in the structure and the considerable surface effects that are present.) Quantum wires are difficult to construct and generally require sophisticated equipment. Quantum dots, however, are fairly readily prepared. Conventional semiconductor techniques can be used to grow small islands the dot on the surface of a semiconductor crystal. Isolated quantum dots are synonymous with nanoparticles and, as such, can be precipitated in glasses or solutions, or prepared as colloids. The quantum dots that have been the subject of most study are of the compounds cadmium sulfide (CdS), zinc sulfide (ZnS), cadmium selenide (CdSe) and zinc selenide (ZnSe). These emit fluorescent light that is a precise function of the dimensions of the quantum dot. For example, CdSe quantum dots of radius 2.9 nm emit at approximately 555 nm, those of radius 3.4 nm emit at approximately 580 nm and those of 4.7 nm radius emit at approximately 625 nm. The colour variation comes about because the band-like properties of the bulk semiconductor are transformed into a closely lying set of discrete energy levels as the dimensions of the particle approach the atomic scale, as described above. The higher group of states, derived from the (nominally empty) conduction band, are, in principle, antibonding or nonbonding orbitals. The lower group of states derived from the (nominally filled) valence band and are, in principle, bonding orbitals. Moreover, the energy gap between the highest orbital in the valence band group, equivalent to a HOMO, drops in energy while the lowest orbital of the conduction band group, equivalent to a LUMO, increases in energy, so that the effective band gap appears to increase steadily as the dot size falls (Figure 10.27a). The photoluminescence, which is relatively pure in colour as the emission spectra are narrow, comes about in the following way. Electrons are excited from the lower set of orbitals to the upper set with ultraviolet radiation, as in the case of ordinary inorganic phosphors (Chapter 9). These excited states subsequently lose energy by nonradiative transitions to end in the lowest orbital of the upper set. Energy is then released as a photon as the electron drops to the topmost orbital of the lower set (Figure 10.27b). Semiconductor nanoparticles (Figure 10.28a) can now be produced with a definite size and narrow size distribution. The relationship between size and band gap allows the photoluminescent colour to be controlled precisely (Figure 10.28b and c). The colour of the photoluminescence will vary with the chemical nature of the nanoparticles, as well as the size of course, so that a wide range of colour tuning is possible, even between the four semiconductors listed above. There are many potential applications for photoluminescent quantum dots, because they constitute minute very bright lamps that can be activated at will by an ultraviolet or blue light probe. Moreover, the colour output is pureinthesensethattheemissionspectrumisnarrow.Theyaremuchbrighterthanthefluorescentdyesdescribed earlier (Section 9.7) and are less easily degraded under normal conditions than dye molecules are. Applications include the biological imaging of processes in living cells, production of quantum dot lasers and white LEDs. These latter devices are made in a similar fashion to that described above, in which a phosphor is coated onto the surface of a blue LED, typically a GaInN device with an output of approximately 460 nm (Section 10.8.4). A layer of CdSe nanoparticles that emit at green, yellow or red wavelengths replaces the phosphor coating. The simple quantum dots described above have a number of shortcomings. The relatively large surface area of the dots reduces the light-generating efficiency considerably. This is in part due to the fact that many of the bonds on the surface atoms are not complete. These dangling bonds serve to trap electrons and holes so that the excited dot loses energy other than by emission of photons. The surface can thus be considered as a defect-rich region that interferes with the mechanism of luminescence. For biological imaging of processes in living cells, not only is the luminous efficiency important, but also the quantum dots must be treated so that they are water soluble; those above are not soluble. Colour and the Optical Properties of Materials 456

antibonding orbitals conduction band

band gap Eg

valence band bonding (a) orbitals

ultraviolet red excitation emission green blue emission emission

(b) ~6 nm ~4 nm ~2 nm

Figure 10.27 Quantum dot colours (schematic). (a) The change in the band gap and band structure of quantum dots as the size falls; (b) Fluorescence colours of different-sized CdSe dots (schematic). Nonradiative transitions shown dotted

The commonest approach to overcoming both of these difficulties is to coat the quantum dot with a thin covering of another material to make core shell structures (Figure 10.29a). For example, a CdSe nanodot core surrounded by a thin shell of ZnS has modified light-emitting properties, while a CdS dot coated with silica (SiO2) or organic surfactants make quantum dots water soluble and less toxic. A curious feature of core shell quantum dots is that they sometimes cease to emit light for a period before returning to normal. This behaviour, termed blinking, is a feature of other fluorescent systems, including fluorescent dyes and GFP.In quantum dots, the cause is believed to be due to the shell acting as an electron trap. When the core of the dot is excited by ultraviolet radiation, an electron hole pair is generated. It appears that if the electron is able to migrate into the shell and become trapped at a surface site, photon emission is suppressed. In effect, the core becomes positively charged; and although the core can still absorb photons, internal energy loss takes over and the excess is lost as heat (Figure 10.29b and c). Eventually, the trapped electron reunites 457 Colour in Metals, Semiconductors and Insulators

r ~2 nm ~4 nm ~6 nm Photoluminescent emission 300 400 500 600 700 800 900 (a) (b) Wavelength / nm

(c) Figure 10.28 Cadmium sulﬁde quantum dots: (a) a CdS dot approximately 8 nm diameter; (b) the photoluminescent colours emitted by CdS quantum dots, schematic; (c) Photoluminescence of two different sized quantum dots (left and centre) and a mixture of the two, giving an approximately white output. [(a) and (c) reproduced with permission from Dr. R. D. Tilley, Victoria University of Wellington, New Zealand] with the core, after which normal photoluminescence is restored. This on off sequence is blinking. Further studies of this phenomenon are currently in progress. Besides roughly spherical dots, many other dot geometries are being created, including rods, dipods, tetrapods and so-called ﬂowers. All are being tested for applications in medicine and biology, photovoltaics, optical computing and other areas.

10.11 Organic Semiconductors and Electroluminescence

10.11.1 Molecular electroluminescence

Organic molecules are generally insulators, with electrons occupying molecular orbitals conﬁned to the spatial region of the molecule itself. The unoccupied, higher energy orbitals are generally antibonding orbitals, and the ﬁlled orbitals are generally bonding orbitals, although, as pointed out earlier, some orbitals are regarded as neutral with respect to bonding and are called nonbonding orbitals. The lowest unoccupied molecular orbital is given the acronym LUMO and the highest energy occupied orbital the acronym HOMO. Organic electroluminescence the generation of light by passing an electric current through an organic solid therefore seems unlikely. However, this is the basis of operation of organic light emitting diodes Colour and the Optical Properties of Materials 458

photon - in + photon out

(a)

photon in no + emission

(b)

photon in no + emission

(c) Figure 10.29 Blinking: (a) under normal conditions core–shell nanodots emit photons efﬁciently; (b) sometimes an electron is trapped on the surface of the shell, preventing emission of a photon; (c) as long as the electron is trapped, photons are absorbed but no light is emitted

(OLEDs) that use organic molecules as the active medium. In principle, electroluminescence comes about in the following way. Under the influence of an applied voltage, electrons are forced into the antibonding orbitals, notionally the LUMO, and electrons are pulled out of the bonding orbitals, notionally the HOMO, which is the same as saying that holes are forced into the HOMO. The electrons and holes recombine and release energy in the form of visible light (Figure 10.30a). The molecules that can be used for this are exactly the same dye molecules that are of use in photoluminescence (Section 9.11). That is, they contain conjugated double bonds that overlap to give delocalized p and p orbitals (Chapter 8). The electrons do not flow through these molecular orbitals in the same way that electrons flow through the band structure of an inorganic semiconductor, but can be imagined to jump or ‘hop’ from one atom to another via the bond structure of the molecule. Each mobile electron causes a small distortion of the surrounding atoms and associated bonds, and the distortion has to be dragged along as the electron hops. This combination of electron plus distortion is called a polaron. Polaron movement requires more energy than the equivalent electron transport in an inorganic semiconductor, and a fairly high voltage must generally be used to obtain electroluminescence in such a system. It has been found that the electrons and holes injected into the organic light-emitting medium do not simply annihilate each other directly, but initially interact to form excitons (Section 10.2). In an exciton the spin of the electron in the HOMO may be antiparallel to the spin of the electron in the LUMO, resulting in singlet state. However, many more of the molecules end up in a triplet state, in which the spin of the electron in the HOMO is parallel to the spin of the electron in the LUMO (Figure 10.30b). Both states may lose energy internally or 459 Colour in Metals, Semiconductors and Insulators

(electrons into LUMO)

photon organic molecular solid out (electrons out = holes into HOMO) (a) +

HOMO

singlet, S = 0

HOMO LUMO

voltage

LUMO HOMO

ground state triplet, S = 1

LUMO

(b) excited state Figure 10.30 Organic electroluminescence. (a) Principle of an organic electroluminescent device; an applied voltage introduces electrons and holes into the material which recombine and emit light. (b) The applied voltage can lead to either a singlet or a triplet excited state of the molecule emit photons. The singlet excited state is more likely to ﬂuoresce and give out electroluminescence. The triplet state has a much longer lifetime, and is more likely to degrade by losing energy internally as heat, but may give out light via phosphorescence (long-lifetime ﬂuorescence) (Figure 10.31). Thus, unless steps are taken to alter this situation, the light output of the system will be fairly low.

10.11.2 Organic light emitting diodes

The transformation of the preceding principles into working devices took place in the period from about 1990. A typical early construction used a thin polymer ﬁlm of the conducting material poly(2-methoxy-5,20-ethyl- hexloxy)-1,4-phenylene vinylene, abbreviated to MEH-PPV (Figure 10.32a). This has an emission maximum close to 625 nm in the orange red portion of the spectrum. In order to introduce electrons into the ﬁlm, a conducting anode and cathode are needed. The energy levels of these two electrodes must match the LUMO at the cathode side and the HOMO at the anode side. The cathode material is calcium. Although this is not an ideal cathode because calcium is highly reactive,the energylevelofthe electrons inthis metal, the Fermi level,is very Colour and the Optical Properties of Materials 460

from anode from cathode

electron - hole pair

singlet exciton triplet exciton

deactivation emission emission deactivation

fast slow

external photons

Figure 10.31 Schematic processes taking place within an organic electroluminescent solid. Note that each step has its own efficiency and that surface processes, important in real devices, are ignored close to that of the LUMO in the polymer and allows easy flow of electrons across the interface (Figure 10.32b). In order for light to leave the device, one electrode must be transparent, and for this the transparent conducting material indium tin oxide (ITO) is used. The Fermi level of this material is similar to the HOMO of the polymer (Figure 10.32b). The completed device (Figure 10.32c) gave out an orange red colour and had an external quantum electroluminescent efficiency Z defined by:

Z ¼ number of photons escaping the device number of charges entering of about 2 %. The early single-layer devices of the sort just described have been changed considerably in the ensuing years and are now replaced by multilayer configurations (Figure 10.33a). Hole or electron injection layers may be introduced next to the anode or cathode to improve the number of mobile charge carriers entering the transport layers. These may adjoin a hole-transporting layer and an electron-transporting layer respectively, which serve to move the charge carriers into the emitting layer. These additional components are not often used at the anode boundary, because the standard anode material, ITO, has an energy band structure that matches the HOMO of many hole-transporting compounds, thus ensuring efficient direct injection of holes into the hole-transporting layer. However, there is more of a problem at the cathode. Calcium is ideal from an energy level viewpoint, but it has a high reactivity in air. A common replacement is magnesium or a calcium-silver or magnesium silver alloy. The Fermi energy of these conductors does not match the LUMO energy of most electron-transporting materials, and it is found that a thin (0.5 nm) layer of lithium fluoride (LiF) as an electron injection layer greatly increases the number of electrons entering the electron-transporting layer. The mechanism by which this enhancement is achieved is not understood. 461 Colour in Metals, Semiconductors and Insulators

MEH - PPV

n O (a)

ITO MEH - PPV calcium anode emitter cathode π* 2.8 eV 2.7 eV

π 4.7 eV (b) 4.9 eV

glass substrate light output

(c) + Figure 10.32 (a) The skeletal structure of poly(2-methoxy-5,20-ethyl-hexloxy)-1,4-phenylene vinylene (MEH-PPV). The segment in brackets is repeated many times in the polymer. (b) Energy-level diagram for the anode, emitting polymer and cathode. (c) Schematic of OLED

Although the transporting layers can be directly coupled to the emitting layer, they are often connected to exciton blocking layers. These latter layers ‘reflect’ diffusing excitons and effectively confine them to the emitter layer (Figure 10.33a). The electroluminescent layer that emits photons may just rely on the energy levels of the HOMO and LUMO, or may contain dopants with a different energy gap. In actual devices, not all of these layers are present, and those that are have been chosen carefully so that the energies of the HOMO and LUMO pairs are matched up (Figure 10.33b). Much effort has been expended in discovering new materials for these devices. An important advancewas the incorporation of heavy metals such as platinum or iridium into the emitting layer. These heavy metals interact strongly with triplet states, so that they emit photons rather than lose energy by nonradiative transitions, thus immediately improving the external quantum efficiency by a factor of three or four. Colour and the Optical Properties of Materials 462

light (a)

anode (+) hole injection layer (HIL) hole generation and transport hole transport layer (HTL) exciton blocking layer (EBL)

light production emitting layers (EL)

exciton blocking layer (EBL) electron generation and transport electron transport layer (ETL) electron injection layer (EIL) cathode (-)

(b) -2 LUMO energy EL HTL EBL Ca:Ag (-) dopant energy levels in EL -3 ETL

-4 ITO (+)

-5 HOMO energy

-6 40 nm 15 nm

-7 25 nm 40 nm Figure 10.33 (a) Generalized schematic diagram of a multilayer OLED. (b) Schematic energy-level diagram of a blue-emitting OLED; the top of each coloured band represents the LUMO energy of that material and the bottom of each coloured band represents the HOMO energy. The Fermi energy of the ITO anode and Ca:Ag cathode are also included. [Adapted with permission from S. Ye et al., ‘Wide-Energy-Gap Host Materials for Blue Phosphorescent Organic Light-Emitting Diodes’, Chem. Mater. 21, 1333–1342. Copyright 2009 American Chemical Society]. HTL ¼ N,N0-di(naphthalene-1-yl)-N,N0-diphenylbenzidene; ETL ¼ 2,9-dimethyl-4,7-diphenyl-1,10-phenanthrolene; EBL ¼ N,N0-dicarbazolyl-3,5-benzene; EL ¼ 1,3-bis(9-phenyl-9H-ﬂuoren-9-yl) benzene host plus Ir(III) bis(40,60-diﬂuorophenylpyridinato)tetrakis(1-pyrazolyl) borate

Apart from films made from small molecules or polymers, molecular dendrites are of increasing importance. Dendrites are ‘bushy’ organic molecules that consist of a core surrounded by branching linear chains of atoms to give a tree-like structure. The core of the dendrite is the luminescent centre. The branches need to transport charge and are often conjugated units. The outer ends of the branches can be modified by attaching a variety of 463 Colour in Metals, Semiconductors and Insulators surface groups to modify the ease of processing of these materials. Many dendrimers that have been explored for use in OLEDs have iridium at the core, thus making use of the ability of this metal to induce luminescence from triplet energy states. White-light-emitting OLEDs can be produced in the same way as described with inorganic LEDs; that is, a blue emitter can be combined with a yellow phosphor. In displays, additive colour mixing can give the impression of white if three separate OLEDs, emitting red, green and blue, combine. With OLEDs it is also possible to combine two or more different dendrimers in the emitting layer to give two separate colour outputs, such as yellow and blue, so as to achieve white light output. In all OLEDs, light extraction is hampered because of the differing refractive indices of the materials, which results in total internal reflection at the many interfaces. This remains a problem. At present the internal quantum efficiency of an OLED is about five times the external quantum efficiency, i.e. only about one in five of the photons generated in the emitting layer leaves the device.

10.12 Electrochromic Films

Electrochromic materials are compounds which change colour reversibly when subjected to an electric field. Colour change in electrochromic films is mediated by oxidation or reduction, usually accompanied by counter- ion transport to maintain charge neutrality. An electrochromic device is thus a form of electrochemical cell. Reduction processes, which are equivalent to a gain in electrons, take place at the cathode. This is also referred to as n-doping. Materials in which the significant colour change is induced by reduction are said to be cathodically coloured. Oxidation processes, which are equivalent to a loss of electrons, take place at the anode. This is also called p-doping. Materials in which the significant colour change is induced by oxidation are said to be anodically coloured. The transparent or colourless form of the electrochromic compound is often called the bleached state. There are two basic types of electrochromic reaction:

coloured > bleached ðiÞ coloured A > coloured B ðiiÞ

From a practical point of view, electrochromic materials are mostly employed in thin-film form as elements in electrochromic displays. The simplest, asymmetric arrangement, is one in which there is a single electrochromic film next to either the anode or the cathode, linked to the counter-electrode by an electrolyte which is also the source and sink for the ions involved in the colour changes. In these devices, an applied voltage can be set up so as to drive ions and electrons into the electrochromic material, changing the colour of the film. Reversal of the voltage drives the ions and electrons in the opposite direction, causing the film to revert to its original state (Figure 10.34a and b). In a dual film arrangement, different electrochromic films are in contact with both electrodes (Figure 10.34c). In this type of device the electrochromic films are coloured in tandem, so as to increase the contrast developed. The electrodes are generally made from the transparent conductor indium tin oxide (ITO). The efficiency of an electrochromic device Z is given by:

D ð = Þ Z ¼ A ¼ log Tox Tred Q Q where DA is the change in absorbance (optical density) produced by an injection or removal of charge Q per unit area of ﬁlm and Tox and Tred are the transmittance of the ﬁlm in the oxidized and reduced states respectively (Section 1.13). Colour and the Optical Properties of Materials 464

glass support

e– ITO cathode electrochromic layer

V electrolyte and source of ions

ITO anode e–

glass support (a)

glass support

e– ITO cathode

electrolyte and source of ions V electrochromic layer ITO anode e–

glass support (b)

glass support

e– ITO cathode electrochromic layer 1

V electrolyte and source of ions electrochromic layer 2 ITO anode e–

glass support (c) Figure 10.34 Electrochromic devices (schematic). (a), (b) Asymmetric cell design in which the electrochromic film is located next to the cathode or the anode. (c) Dual-film device in which electrochromic films are located adjacent to both electrodes. ITO represents the transparent conductor indium tin oxide

The speed of darkening of a film, or its opposite, bleaching, depends mainly on ionic diffusion in a weak electric field. At ordinary temperatures this is too slow for fast displays such as television, but it is satisfactory for electronic notice boards or shop signs and similar displays that are relatively permanent. Recently, electrochromic films have been explored for use as dynamic camouflage, involving switching between green and brown tones. However, the best-known application is in ‘smart’ windows or mirrors that control the amount of light reflected or transmitted. 465 Colour in Metals, Semiconductors and Insulators

(a) (b)

glass support

e– e– ITO cathode

WO3 electrochromic layer + M electrolyte V

M M+ source / sink e– ITO anode e–

glass support (c)

Figure 10.35 Inorganic electrochromic ﬁlms. (a) The idealized structure of tungsten trioxide (WO3) composed of corner-sharing WO6 octahedra. (b) The idealized perovskite tungsten bronze structure, AxWO3, in which large A cations are interpolated into the cages in the parent WO3 phase. (c) An electrochromic device schematic using a thin ﬁlm of WO3 as the electrochromic phase

10.12.1 Tungsten trioxide electrochromic ﬁlms

The material most widely explored for electrochromic devices is tungsten trioxide (WO3), which in the bulk is pale yellow and an insulator (see Figure 10.1). Thin films are transparent. The crystal structure of this oxide is rather open and builtof corner-linked WO6 octahedra (Figure 10.35a). Electrochromic films are coloured by the formation of tungsten bronzes, MxWO3. There are a number of tungsten bronze structures, but the ones utilized for electrochromic purposes are the perovskite bronzes, in which cations such as Li, Na or K occupy cage sites between the corner-linked WO6 octahedra of the parent phase (Figure 10.35b). The colour of these is dark blue black. The phase range over which the perovskite bronze structure is stable is greatest for the Li phases and smallest for the K phases. The hydrogen bronzes, HxWO3 are also deeply coloured blue black, although these are rather different from the alkali metal phases and are probably best regarded as a nonstoichiometric hydroxide, WO3 x(OH)y. Although the colour of the tungsten bronzes has not been explained fully over all of the composition range, at the low concentrations employed in electrochromic films the blue black colour induced is so similar to that of reduced tungsten trioxide that it is presumed that charge transfer between two valence states of tungsten is occurring. If so, colour may then be attributed to W5þ W6þ or W4þ W6þ couples. The principle of an electrochromic device using tungsten trioxide films is not too difficult to envisage. It is necessary to drive some appropriate metal, such as Li, into the WO3 film using an applied voltage. This will make the tungsten trioxide turn into a blue black tungsten bronze. Reversal of the voltage must remove the interpolated metal and regenerate the colourless state. The reaction can be schematically written as: Colour and the Optical Properties of Materials 466

þ WO3 þ xM þ xe > MxWO3 transparent ðbleachedÞ film þ electrons > dark film

In this material, the transparent (oxidized) film is transformed into the coloured (reduced) film by the gain of electrons (that is, reduction or n-doping) and WO3 is a cathodically coloured substance. Devices are constructed as a series of thin films on glass. Transparent conducting electrodes, usually ITO,sandwich a film of þ þ WO3,anM ion conducting electrolyte and a separate reservoir (source/sink) of metal M ions if needed (Figure 10.35c). Dark electrochromic films making use of sodium tungsten bronzes, NaxWO3, have been obtained using the nonstoichiometric fast-ion conductor b-alumina (Na1 þ xAl11O17 þ x/2). This compound has a broad composition range, with x taking values between 0.15 and 0.3, and is an excellent conductor for Naþ ions so that it can þ act as both the source and sink for these ions. The power supply can force Na ions to migrate into the WO3 to form a dark NaxWO3 bronze, or remove them back into the b-alumina reservoir to turn the bronze back into colourless WO3 (Figure 10.36a).

– + – e e ITO cathode +– WO3 + xNa + x e → Nax WO 3 + → x + Na Na1+x Al 11 O 17+x/2 NaAl 11 O 17 + Na + x/2 O2 – e ITO anode (a) _ e–

– + – e e ITO cathode +– WO3 + xLi + x e → Lix WO3 Li+ gel electrolyte containing Li salt → xLi+– + xe – e ITO anode (b) _ e–

– + – e e ITO cathode +– WO3 + xH + x e → HWOx 3 H+ HUP proton conductor +– – H2O ITO anode: x/2 H O → xH + x/2 O2 + xe _ e– e 2 (c) moist air

– + – e e ITO cathode +– WO3 + xH + x e → HWOx 3 H+ HUP proton conductor +– NiOyz H → NiO yz–x H + xH + xe

e– ITO anode (d) _ e–

Figure 10.36 Electrochromic devices using tungsten trioxide (schematic). (a) Colour due to the formation of sodium tungsten bronze, NaxWO3. (b) Colour due to the formation of lithium tungsten bronze, LixWO3. (c) Colour due to the formation of hydrogen tungsten bronze, WO3x(OH)y. (d) Colour due to the formation of hydrogen tungsten bronze and oxidized nickel oxy-hydroxide 467 Colour in Metals, Semiconductors and Insulators

Lithium can be inserted into the WO3 thin ﬁlm if a lithium reservoir is substituted for a sodium source. A material that can be used for this purpose is an electrolyte consisting of a gel containing a readily ionized lithium salt. Application of an electric ﬁeld can drive Liþ out of or back into the gel reservoir at will (Figure 10.36b). In the case of hydrogen tungsten bronzes, HxWO3, it is possible to use the decomposition of water vapour in the atmosphere as a source of Hþ . Decomposition takes place on the outer ITO electrode:

þ 2H2O ! O2 ðgÞþ4H þ 4e

This electrochemical decomposition requires about 1 V at the electrode surface. To drive the protons into the WO3 ﬁlm, a proton-conducting electrolyte, typically hydrogen uranyl phosphate, HUO2PO44H2O (HUP), is utilized. The Hþ produced can pass through the proton-conducting electrolyte to form the bronze, using electrons from the other electrode (Figure 10.36c). On reversal of the applied voltage the Hþ ions are pulled out from the bronze and the ﬁlm becomes colourless once more.

10.12.2 Inorganic electrochromic materials

Apart from tungsten trioxide, a number of other inorganic materials show electrochromic behaviour. Among the most important of these are hydrated nickel oxide, niobium pentoxide and Prussian blue. Hydrated nickel oxide (Ni(II)OxHy,), a poorly characterized material, is pale green in bulk and transparent in thin-ﬁlm form. It is readily converted to a metastable brown oxyhydroxide containing Ni(III). The complex chemistry of the reversible reaction can be approximated as:

þ NiOxHy > NiOxHy z þ zH þ ze transparent ðbleachedÞ film > dark film þ electrons

In this material, the transparent (reduced) film is transformed into the coloured (oxidised) film by the loss of electrons (that is, oxidation or p-doping) and hydrated nickel oxide is an anodically coloured substance. Hydrated nickel oxide can be used in conjunction with tungsten trioxide films, enhancing the darkening effect of the tungsten bronze layer and so improving the darkening characteristics of the device (Figure 10.36d). Niobium pentoxide (Nb2O5), which is colourless and forms transparent films, has a structure related to that of tungsten trioxide and, like this latter material, can take in Liþ or Hþ to form a dark blue black phase on reduction. The reaction can be written:

þ Nb2O5 þ xM þ xe > MxNb2O5 transparent ðbleachedÞ film þ electrons > dark film

In this material, the transparent (oxidized) film is transformed into the coloured (reduced) film by the gain of electrons; that is, reduction or n-doping, and Nb2O5 is a cathodically coloured substance analogous to WO3. 3þ 2þ 2þ 2þ Dark blue Prussian blue, KFe Fe (CN)6, is also readily oxidized to colourless K2Fe Fe (CN)6, Prussian white (see also Section 8.10). Thus thin films of Prussian blue can be made transparent by the reduction reaction and the transparent films darkened by oxidation. The essence of the reaction is the interconversion of Fe2þ and Fe3þ :

Fe2 þ Fe2 þ > Fe3 þ Fe2 þ þ e transparent ðbleachedÞ film > dark film þ electrons Colour and the Optical Properties of Materials 468

In this material, the transparent (reduced) ﬁlm is transformed into the coloured (oxidized) ﬁlm by the loss of electrons (that is, oxidation or p-doping) and Prussian white is an anodically coloured substance. An electrochromic cell using both WO3 and Prussian white to enhance contrast can be fabricated, operating similarly to that with WO3 and hydrated nickel oxide described above.

10.12.3 Electrochromic molecules

Many organic molecules can be coloured via oxidation or reduction, making them ideal targets for electrochromic devices. In this section, organic conjugated polymers, a widely explored group, will be considered as illustrative. The most important of these polymers are those that are also electrically conducting polymers, notably poly(pyrrole) (PPy), poly(thiophene) (PTh) and poly(aniline) (PANI) and derivatives of these parent phases. The colours of these polymers derives from a p p (HOMO LUMO) transition. In the normal insulating (or neutral) state these materials are semiconductors, with a band gap Eg defined by the separation between the HOMO and the LUMO which also controls the colour of the neutral state. Narrow band gap polymers will be coloured, whereas broad band gap polymers will be transparent. Oxidation or reduction (p-doping or n-doping, depending upon the polymer) accompanied by ion insertion or removal turns these materials into electrical conductors. The conducting phases show absorption bands due to transitions involving the newly created charge carriers, generally thought to be polarons (Section 10.11). As the material is gradually oxidised or reduced, colour due to the original p p transitions diminishes and new transitions to the modified band structure and transitions involving the newly formed polarons and bipolarons change the colour of the polymer. As with inorganic materials, some electrochromic polymers may be more readily subject to reduction, in which case they are cathodically coloured. These materials generally have a relatively small band gap, of the order of 1.7 eV, and so tend to be coloured in the neutral (insulating) state. Other electrochromic polymers are more readily subject to oxidation, in which case they are anodically coloured. These materials tend to have a large band gap, of the order of 2.5 eV, and so are usually transparent in the neutral (insulating) state. An advantage of organic molecules over inorganic materials is that the colours available can be modified by inserting substituents into the structure using well-known organic chemistry methodology. This leads to a third type of electrochromic reaction:

coloured A > coloured B > coloured C ðiiiÞ

This last group, in which several different coloured forms can be cycled, can be found in some polymers as the degree of doping changes, or may be manufactured from copolymers of monomers showing just two colour states.

10.12.4 Electrochromic polymers

From the large number of polymeric electrochromic materials so far investigated, two are illustrated here as representative, PANI and the alkoxy-substituted polythiophene poly(3,4-ethylenedioxythiophene) (PEDOT). PANI, derived from the small molecule aniline (Figure 10.37a), exists in three basic forms, each of which shows a different colour; the polymer, therefore, is polychromic. The polymers can be regarded as built up from two end-species: aromatic, reduced, leucoemeraldine, yellow, and clear in thin -ﬁlm form; and quinoid, oxidized, blue violet, pernigraniline. The 1:1 intergrowth of these two structures is the green blue emeraldine (Figure 10.37b). 469 Colour in Metals, Semiconductors and Insulators

H N aniline (a) H

N N N N polyaniline (PANI) H H a b (b) n

An- An- - - H + H + An An N N N+ N+ polyaniline (PANI) salt H H HH a b (c) n

leucoemeraldine yellow, a = 1, b = 0, reduced

emeraldine, green / blue, a = b = 1/2, neutral

pernigraniline, blue / violet, a = 0, b = 1, oxidized

Figure 10.37 The idealised structures of polyaniline (PANI): (a) aniline; (b) leucoemeraldine, a ¼ 1, b ¼ 0; emeraldine, a ¼ b ¼ 1/2; pernigraniline, a ¼ 0, b ¼ 1; (c) polyaniline salts, in which the anions, An and cations or þ H are combined with the polymer

This simple description masks a complex polymer chemistry and physics. Generally, PANI is prepared in a doped conducting emeraldine salt form ES-I. The dopant is often a simple acid such as HCl, and one or both parts of the polymer can be modified in this way (Figure 10.37c). The acid can be removed to yield an insulting emeraldine base form, EB-I. The material can also be prepared in a different insulating emeraldine base form, EB-II, which can be made conducting by doping with, for example, HCl to form the conducting emeraldine salt ES-II. All of these products show variable degrees of crystallinity. For electrochromic device use, these materials are often dissolved in a solvent and cast as films, which are partly crystalline and show conducting or insulating behaviour depending upon whether they are doped or not. In addition, the colour changes slightly on doping, so that ES forms are green and EB forms blue. In order to improve the device characteristics and performance, the PANI films are frequently reacted with other dopants, such as D,L-camphor-10-sulfonic acid (CSA) or poly(styrene sulfonic acid) (PSS) to give PANI CSA or PANI PSS for instance. In this group of materials, the colour of the film can be cycled between pale yellow (leucoemeraldine) and dark blue (pernigraniline). PANI in these devices is anodically coloured, as the reduced transparent form is oxidized to the coloured form. The widely used electrochromic polymer PEDOT is derived from polythiophene (Figure 10.38) and has two coloured states: red and blue. Like PANI and its derivatives, polythiophene and derivatives also have an aromatic-type and a quinoid-type structure, but in this case the quinoid type is of higher energy and not found in normal preparations. PEDOT itself shows two colour varieties: a transparent oxidized form and a reduced blue form. The band gap of the material is about 1.78 eVand, like WO3, becomes coloured on reduction, i.e. it is cathodically coloured. Devices using polymers are constructed in a similar way to those using inorganic materials. They can contain one active electrochromic polymer layer, one electrochromic polymer layer coupled with an inorganic Colour and the Optical Properties of Materials 470

O O

S (a) S (b) n (c) S n

Figure 10.38 The structures of (a) thiophene, (b) polythiophene and (c) poly(3,4-ethylenedioxythiophene (PEDOT) material, or two organic polymer layers. As before, the electrochromic layers are separated by an electrolyte layer, often in gel form, containing the ions needed to maintain charge balance. A cell made from a cathodic ﬁlm of PEDOTand an anodic ﬁlm of PANI PSS is an example (Figure 10.39). The anode of the cell, where the reduced form of the electrochromic polymer is to be reduced to a coloured form, is in contact with PANI PSS. The cathode of the cell, where the oxidized form of the electrochromic polymer is reduced to a coloured form, þ is in contact with PEDOT PSS. The electrolyte contains lithium chlorate, giving Li and ClO4 ions, in a gel matrix. The schematic reactions taking place are as follows: at the anode

> ð þ Þþ PANI-- PSS PANI-- PSS ClO4 e pale yellow reduced form > blue oxidized form at the cathode

PEDOT-- PSS þ e þ Li þ > ðLi þ PEDOT-- PSSÞ pale blue oxidized form > dark blue reduced form

In these reactions, both electrochromic ﬁlms darken simultaneously, to give an overall transparent to blue electrochromic change. Many polythiophenes also show thermochromism. The reasons are related to the occurrence of photochromic behaviour. The polythiophene molecules are usually planar when cold, have considerable electron delocalisation and a smaller band gap. The ‘cold’ colour tends to be red. As the temperature increases, the backbone of the polymer can buckle or twist. This inhibits the amount of electron delocalisation and has the effect of increasing the band gap of the molecule. Thus, the colour change is from red towards the green blue as the temperature increases.

ITO cathode – – e e PEDOT–PSS: pale blue oxidised → dark blue reduced + Li – LiClO containing gel electrolyte ClO4 4 PANI–PSS: yellow reduced→ blue oxidised

e– ITO anode e– Figure 10.39 Schematic electrochromic device utilising PEDOT cathodic and PANI anodic electrochromic ﬁlms 471 Colour in Metals, Semiconductors and Insulators

10.13 Photovoltaics

10.13.1 Photoconductivity and photovoltaic solar cells

If radiation of a suitable wavelength falls on a semiconductor, it will excite electrons across the band gap. One result is that a voltage develops across the semiconductor and the conductivity of the material increases. Materials that show a voltage on illumination are called photovoltaic materials. The magnitude of the effect is roughly proportional to the light intensity. These properties, called the photovoltaic effect or photoconductive effect, have been used in light meters, exposure meters and automatic shutters in cameras, and many other devices. The first exposure meters for the measurement of light amounts available for photography used selenium (Se), cadmium sulfide (CdS) or silicon (Si). In the case of selenium, the photovoltage is large enough to be measured directly and converted to an exposure value. Cadmium sulfide and silicon need voltage amplification, and these materials need a power source, usually a battery, to give a reading. A DC voltage applied to the ends of a semiconductor will also allow the photoeffect to be measured. The increase in conductivity on illumination provides a means of measurement of the amount of incident light falling on the device. A p n junction can act in a similar way to a single piece of semiconductor. However, the control afforded by the junction makes the device, called a photodiode, far more flexible; as a result, photodiodes are widely used, especially in solar cells. A solar cell is specialist large-area p n junction with a depletion region approximately 500 nm thick. (Solar cells must have a large area, to collect as much sunlight as possible.) In addition, the normal built-in potential that exists across the junction, due to the space charge, is engineered to be high (Figure 10.40a). The junction is not connected to any external power source. Holes and electrons produced in

photon

n-type material junction region space charge region

p-type material Ip hole (a) electron

sunlight incident on anti-reflective coating

negative front contact _ n-type semiconductor ~ 10–6 m

p-type semiconductor + reflecting layer (b) positive back contact Figure 10.40 Solar cell schematics. (a) Sunlight incident upon a p–n diode junction creates electron–hole pairs that are swept into the external circuit by the built-in field in the junction region. (b) An operating cell needs an antireflection front coating, a junction region near to the illuminated surface and a back reflecting layer to optimise cell efficiency Colour and the Optical Properties of Materials 472 the junction region by sunlight are swept across the depletion region by the high built-in space charge present, the electrons going from the p to n region and the holes from the n to p region. This process, called drift, charges the p region more positive and the n region more negative, and produces a photocurrent current Ip across the junction which generates a photovoltage. The photovoltage corresponds to a forward bias, and so will cause a current I toflow. At equilibrium I ¼ Ip. Should anexternal load R be connected, some current can flow through it, and so do useful work. Not all electron hole pairs are created in the junction region. Those generated outside it must diffuse through the solid until they approach the junction, where they are subjected to the internal field and can contribute to the photocurrent. This simple description obscures the amount of effort required to produce an efficient solar cell. The simplest amendments are the requirement of an antireflection surface to maximize the number of photons reaching the semiconductor, a thin initial semiconductor layer, so that the optimum number of photons reach the depletion region, and an underlying reflective layer that redirects any photons that pass straight through the lower semiconductor layer back towards the junction region (Figure 10.40b). A number of basic considerations influence the design of solar cells. For example, because impurities and defects trap mobile electron and holes, which greatly reduces the efficiency of the cell, high-purity materials are mandatory, although this increases cell costs considerably. It is also clear that the band gap of the solar cell materials chosen must utilize as much of the wavelength spread available in sunlight (approximately 350 2500 nm, 3.5 0.5 eV,with a peak in the yellow green at 550 nm) as possible. Moreover, indirect band gap materials have a lower efficiency than direct band gap materials. The cells that currently (2010) show the highest efficiency are based upon silicon. A drawback is that silicon is an indirect band gap solid and does not absorb across all of the desired energy range very efficiently. In an effort to overcome this problem, amorphous silicon, which behaves as a direct band gap material, is used in many devices. In order to increase efficiency over that currently available, other cell materials investigated include the semiconductors cadmium telluride (CdTe), cadmium sulfide (CdS), copper indium selenide (CuInSe2) and mixed copper selenides (Cu(In,Ga)Se2) and quantum dots (Section 10.10). Recently, much effort has been put into the construction of solar cells using polymers. These have the great advantages of low weight and flexibility. However, efficiencies are not yet adequate for commercial purposes. Solar concentrators, mirrors or lenses that focus the sunlight onto the photoactive layers, are widely used to increase efficiency, as are mobile systems that are able to follow the motion of the sun throughout the day. To absorb as much as possible of the high-irradiance part of the solar spectrum, cells have been stacked in series; for example, GaInP2, GaAs and Ge, which is able to utilise photons from 590 to 1200 nm, 2.1 to 1.0 eV.

10.13.2 Dye-sensitised solar cells

In a conventional solar cell, the conversion of the light to free charge carriers is carried out by the solid semiconductor, which then has to move these away from the junction in order to obtain energy. To achieve good efﬁciency the photons need to be absorbed close to the p n junction. Electron hole pairs created elsewhere have to diffuse to the junction region and, unless the materials are of high purity, recombination is likely. The method of conversion of sunlight to energy of most importance on the Earth, photosynthesis, uses slightly different methods of achieving the same objective. The central reactions are oxidation and reduction. Photoelectrochemical cells, of which dye-sensitised solar cells (also called Gratzel€ cells) are an important example, aim to mimic this process. The task of harvesting the light is left to a sensitiser, which is a dye molecule, and the carrier transport task is allocated to a semiconductor. Because the charge separation takes place in the dye, the purity and defect structure of the semiconductor are not crucial to satisfactory operation. 473 Colour in Metals, Semiconductors and Insulators

current load

hole electron R R – e R oxidized species photon – – R R– R reduced species TCO TCO anode cathode TCO transparent dye electrolyte (a) semiconductor conducting oxide

CB conduction band CB S* FL VB valence band V FL Fermi level RO VB (b) S RO redox potential Figure 10.41 Dye-sensitised solar cell schematics. (a) Sunlight absorbed by the dye liberates an electron into the semiconductor. The dye is regenerated by interaction with an internal redox couple. (b) Schematic energy-level diagram. The cell voltage V is the difference between the Fermi level of the semiconductor and the redox potential of the couple

The reactions in the cell are (Figure 10.41a): 1. Excitation of the sensitiser dye S by a photon:

S þ hn ! S*

2. The excited sensitiser S loses an electron, which moves into the conduction band of the semiconductor:

S* ! S þ þ e ðsemiconductorÞ

3. The electron moves through the conduction band of the semiconductor to the transparent conducting anode, also called the working electrode, which acts as the electron collector. Thereafter, electrons traverse the external circuit to arrive at the cathode, also called the counter electrode. 4. Electrons arriving at the counter electrode reduce a redox couple, R/R , usually in a liquid electrolyte:

RðaqÞþe ! R ðaqÞ

where (aq) represents an aqueous solvent. 5. The sensitiser is regenerated by reaction with the reduced half of the redox couple:

S þ þ R ! S þ R

A large number of different dyes have been tried in the role of sensitiser in conjunction with a variety of inorganic oxides, including ZnO and Nb2O5 as the semiconductor. Currently (2010), the best efﬁciency is Colour and the Optical Properties of Materials 474 obtained with cells using dye molecules containing Ru(II), such as cis-dithiocyanatobis-2,20-bipyridine-4- 0 COOH, 4 -COO Ru(II), in combination with nanocrystalline anatase (titanium dioxide, TiO2) as the semiconductor. The charge states on the dye correspond formally to the conversion of Ru(II) to Ru(III) via photon interaction:

RuðIIÞþhn ! RuðIIIÞþe

The dye is absorbed onto the surfaces of the anatase crystallites to give a large surface area whilst maintaining compact electrode geometry. The transparent conducting oxide electrodes are usually tin oxide doped with ﬂuorine (SnO2:F). The redox couple usually chosen is iodide triiodide in solution:

I3 ðaqÞþ2e ! 3I ðaqÞ

In order to catalyse the oxidation-reduction equilibria in the electrolyte, the cathode electrode is coated with a thin layer of platinum. The cell output depends upon the relative positions of the energy levels in the adjoining components, which must be matched for optimum efﬁciency. In the present cell design, this is given by the difference between the redox potential of the oxidation reduction couple chosen and the Fermi level of the semiconductor (Figure 10.41b). For the triiodide iodide couple the redox potential is þ 0.54 V. The Fermi level of the nanostructured anatase is about 0.4 V, so that the cell voltage is approximately 0.54 V þ 0.4 V, i.e. 0.94 V. There is much current research directed towards improving the dyes used in these cells. The stability of the dye above is limited by the thiocyanate (SCN) ligands within it, and dyes which do not contain these groups are under active consideration as alternatives. The desirability of having a stable organic dye that does not contain a heavy metal is also an important research goal. Similarly, there is interest in replacing the expensive platinum cathode with an organic conductor such as PEDOT. Active research in this area means that new cell speciﬁcations are continually appearing in the literature.

10.14 Digital Photography

10.14.1 Charge coupled devices

In the space of a few years, digital photography has become the standard recording technique for amateur photographers, almost totally replacing film photography. Some time before this, digital imaging replaced photographic film as an image storage medium in many areas of scientific and medical research, initially starting with astronomy. The difference between digital and conventional photography is simply in the way in which the information content of light is captured and stored, photographic film (Section 10.16) versus (mainly) charge coupled devices (CCDs). The concept of the CCD was proposed by Boyle and Smith in 1969, as a contender for computer memory.8 Charges were localised in small volumes (bubbles) in a silicon chip. Each bubble could represent a 0 or 1, depending upon the presence or absence of charge, and bubbles could be annihilated or moved around by changing the voltage applied to an array of electrodes covering the silicon slice. The name ‘charge coupled device’ springs from the mechanism by which the bubbles were moved in concert. Although the memory storage aspect did not result in commercial applications, it was apparent at the time that CCDs had other

8 This was an attempt to mimic another related data storage technology, the magnetic bubble memory. Magnetic bubble memories never became a commercial success as they were overtaken by other means of data storage, especially optical data storage. 475 Colour in Metals, Semiconductors and Insulators

heavily doped poly-silicon gate silicon dioxide

p-type silicon

(a)

V + V ++ V +++

(b)

+ ++++

----- (c)

Figure 10.42 CCDs: (a) schematic MOS pixel; (b) potential well formation under a gate voltage; (c) steady-state capacitor-like charge distribution in an MOS pixel applications. Among the first was as a light-collecting alternative to photographic film. The first commercial image recording sensor was in place at the Kitt Peak Observatory in 1979, just 10 years after the initial concept. CCDs employ similar basics to photodiodes (Section 10.13). Photons falling upon a silicon slice generate electron hole pairs. The number of electron hole pairs that are produced during an exposure is measured and form the datum that ends up as a pixel of the image. The structure of a pixel consists of a photoactive layer of p-type silicon. This is covered with a thin layer of silicon dioxide (SiO2) and this is topped with a conductor, originally aluminium, but now heavily doped polycrystalline silicon, to form a metal oxide semiconductor (MOS) device (Figure 10.42a). The polycrystalline silicon (M) layer is called the gate. A positive voltage applied to the gate will create an electric field across the thin layer of insulator, the O layer, mimicking the action of a parallel-plate capacitor. This field repels holes in the adjacent volume of the p-type semiconductor, the S layer, which creates a depletion region similar to that in a p n junction diode. This region acts as a potential well to trap charges, and so can be thought of as a pocket or bucket in which charge is stored. As the gate voltage increases, the field increases and extends the region where the holes are repelled, increasing the size of the pocket (Figure 10.42b). A number of simultaneous processes occur in the pixel. 1. Electron hole pairs are continually created throughout the p-type silicon slab by thermal energy and by incident light photons. The electrons that form in the bulk of the p-type region have a short lifetime before they recombine with the excess of holes present. However, those that are formed in the potential well under the gate have a much longer lifetime because the hole population there is low. The field present sweeps the electrons towards the oxide layer. However, this build up of charge gradually diminishes the field present. 2. Thermal diffusion causes some holes to enter the depletion volume despite the electric field. These will be annihilated by the electrons present. 3. As more electrons accumulate at the oxide interface, the field that maintains the potential well weakens, allowing more and more holes to diffuse in. Ultimately, a dynamic equilibrium is reached. Colour and the Optical Properties of Materials 476

4. The amount of charge accumulated in the potential well then depends upon the rate of creation of thermal electron hole pairs, the rate of creation of photogenerated electron hole pairs and the rate of destruction of electrons by hole diffusion. Provided that the rate of photogeneration is much greater than that of the competing processes, charge will build up in the pocket in direct proportion to the amount of light incident upon the pixel. 5. At this stage, there is a charge separation across the oxide layer that resembles that in a charged capacitor (Figure 10.42c). As with a capacitor, if the initial gate voltage is removed, then the charges will remain in place, or, more correctly, will slowly leak away. The amount of charge is measured before this happens. The potential well is emptied and the next exposure can be recorded. (The sophisticated mechanisms by which these records are read out and stored can be explored via the sources in this chapter’s Further Reading.)

10.14.2 CCD photography

There are a number of features of CCDs that are of importance in photography. The first to note is that the CCD recording device is linear. That is, the charge that accumulates is directly proportional to the light irradiance, so that the data recorded in a pixel is proportional to the light irradiance. This is not true of photographic film, and represents an improvement over the older technique, especially in scientific recording. A related advantage is that the spectral range over which a silicon-based CCD is sensitive is far greater than that of photographic film. Silicon can detect well into the infrared, although these wavelengths are not visible to the eye. Naturally, there is a limit as to how much charge can be accumulated inanypixel. This is the full-well capacity of the sensor, which is a function of the temperature of the device, the doping levels and the physical dimensions of the various parts. After the full-well capacity has been reached during an exposure, no further information is registered. The dynamic range of a sensor, therefore, is directly associated with the full-well capacity. The rate of thermal generation of electron hole pairs is an important parameter. In low light level use, such as astronomy, where long exposure times are needed to obtain an image of a faint object, the thermal ‘dark current’ adds noise to the information. In these cases, cooling the detector with liquid nitrogen is employed. For day-to day photography this background noise is not usually of importance. Although CCDs are linear in their response, they do not respond evenly to all wavelengths unless some modifications are made to the basic structure described above. This is because incident photons have to traverse the gate electrode before reaching the photoactive silicon layer. Electron hole pairs and other energy- absorbing processes take place in the gate layer, and higher energy photons lose energy faster than lower energy photons. This is measured by the linear absorption coefficient of silicon, which shows that photons at the violet end of the spectrum frequently cannot penetrate the gate electrode. This means that the device is blind to the blue end of the spectrum and only records at the red end. (This is the exact opposite of native silver halide emulsion films; Section 10.16.) There are a number of ways in which this is corrected. The gate layer must be made as thin as possible, allowing the violet end of the spectrum to penetrate. Carefully positioned perforations can be made in the gate to allow violet photons to reach the photoactive silicon layer. The device can also be inverted, so that photons enter the photoactive silicon directly, not through the ‘front’ gate electrode layer. All these are made use of in current cameras. The sensitivity of each pixel, even when the greater absorption of the violet end of the spectrum is balanced, does not match the sensitivity of the eye. To this extent the pixels measure in radiometric units, while the eye responds to photometric units. This sensitivity can be adjusted by using filters. The simplest method is to use a Bayer filter. Each pixel in the array is covered by a filter that only allows one light frequency range through. In order to match the sensitivity of the eye (Section 1.10) these are in a ratio of one red, one blue and two green filters. The disadvantage of filters of any kind is that they significantly reduce the amount of light reaching the photoactive region of the pixel, thus reducing sensitivity considerably. A better (and more costly) technique is 477 Colour in Metals, Semiconductors and Insulators to use three separate CCDs. The image is passed through a prism or dichroic crystal so that the red, green and blue parts of the image are received by a separate sensor array. There are many other techniques for image acquisition that are used in specialist areas, ranging from deep- space astronomy to light microscopy. These can be explored via this chapter’s Further Reading.

10.15 The Colours of Metals

Metals are defined as materials in which the uppermost energy band is only partly filled. This can be imagined to be the logical outcome of shrinking the band gap of a semiconductor to zero. The highest energy attained by electrons in the resulting single band is called the Fermi energy or the Fermi level in a one-dimensional situation. More correctly, this is known as the Fermi surface in the three-dimensional crystal. The key point about a metal is that the higher empty electronic energy levels of a metal are so close to the uppermost filled levels that they form an essentially continuous band of allowed energies. Above the Fermi energy almost all the levels are empty (at absolute zero they are all empty) and so can accept electrons excited from lower energy levels. To a first approximation this means that all incident radiation can be absorbed, irrespective of its wavelength. Intuitively, this would lead one to expect that a metal should appear black. However, each excited electron can immediately fall back to the state that it came from at once, emitting exactly the same energy, causing a flat piece of metal to appear reflective. Ordinary mirrors are metal films deposited onto glass. In a good mirror the absorption and reflection should be identical over thewhole of the spectrum and all colours accurately reflected. Exactly the same absorption and emission processes lead to finely powdered metals having a black appearance. This is because the re-emitted (i.e. ‘reflected’) photons are reabsorbed again in nearby grains and ultimately do not emerge at the ‘angle of reflection’ and so do not enter the eye. To take this absorption into account, the refractive index N of a metal is written as:

N ¼ n þ ik where n is the ‘normal’ refractive index defined in Chapter 2, k is the extinction coefficient, coefficient of absorption or attenuation coefficient, and i is the square root of 1 (Section 2.1). Thevalues of n and k are often called the optical constants of a material, although they vary considerably with the wavelength of the irradiation used as a probe and are not constant at all. For a metal the extinction coefficient k and the refractive index n are both strongly wavelength dependent (Table 10.3). The reflectivity of a metal depends upon n, k and

Table 10.3 Reﬂectivity of copper, silver and golda

Copper Silver Gold Wavelength/nm nkRnkRnkR

705 0.21 4.205 0.956 0.04 4.838 0.993 0.13 4.103 0.971 660 0.22 3.747 0.943 0.05 4.483 0.991 0.14 3.697 0.963 620 0.30 3.206 0.900 0.06 4.152 0.987 0.21 3.272 0.931 549 1.02 2.577 0.619 0.06 3.586 0.983 0.43 2.455 0.787 496 1.22 2.564 0.576 0.05 3.093 0.981 1.04 1.833 0.447 451 1.24 2.397 0.539 0.04 2.657 0.980 1.38 1.914 0.408 397 1.32 2.116 0.464 0.05 2.070 0.963 1.47 1.952 0.407 a Data from: P. B. Johnson, R. W. Christy, Phys. Rev.B6, 4370–4379 (1972). Colour and the Optical Properties of Materials 478 the polarisation of the light. For light falling perpendicularly on a metal surface, polarisation can be ignored and the reﬂectivity is given by:

ðn1Þ2 þ k2 R ¼ ðn þ 1Þ2 þ k2

If k is omitted, the formula reduces to that for a normal insulator such as glass. The colours of copper and gold are due to the fact that the absorption and emission of photons are noticeably dependent on wavelength across the visible (Table 10.3). These data indicate that both gold and copper have rather low reflectivity at the short wavelength end of the spectrum and so yellow and red will consequently be reflected to a greater degree. This leads to the colours observed. Silver, on the other hand, has a high reflectivity which does not vary significantly with wavelength, making it suitable for use in mirrors for astronomical telescopes. It has now largely been replaced in this use by aluminium, which has a similar high and uniform reflectivity over the visible spectrum and which forms a protective transparent oxide film over the metal on exposure to air. Silver films, on the other hand, gradually degrade, especially in polluted atmospheres. Thin flakes of a ductile metal such as aluminium, produced by ball milling, are added to paints to obtain a ‘metallic’ effect. Aluminium is especially suitable from this point of view and is the commonest metal used, but bronzes and copper alloys are also employed for this purpose. Metal flakes are rarely employed alone, usually being used in conjunction with other pigments to produce a shining and attractive finish. A number of compounds, notably metal oxides, change from metallic to insulating behaviour at a definite transition temperature. This is the case, for example, with the oxide VO2. At room temperature this oxide behaves like a poor semiconductor. Above 68 C it becomes a metal, with characteristic reflectivity. This example of thermochromism is brought about by a change of bonding and, hence, of symmetry of the structure, from monoclinic at low temperatures to tetragonal at high temperatures. This type of transition obviously has a value for the fabrication of ‘smart’ windows and similar devices, which can reflect sunlight when the day is hot yet allow it in when the day is cool. The transition temperature is too high for this to be effective in normal climatic conditions, but doping VO2, particularly with WO2 to form V1 xWxO2, with x equal to 1 or 2 at.%, reduces the transition temperature to nearer normal room temperatures.

10.16 The Colours of Metal Nanoparticles

10.16.1 Plasmons

As described in Chapter 5, the optical properties of small metal particles, often referred to as metal sols, colloids or nanoparticles, is dominated by absorption. Mie scattering theory is well able to describe the production of colour of spherical particles by a combination of scattering and absorption, but it does not address the origin of the absorption itself. The earliest theories to attempt this were developed in the years between 1899 and 1903 by Drude and then Lorentz. The collectivetheory that emerged is known as the Drude Lorentz free electron theory of metals. It is remarkably successful and, besides acting as the nucleation site for many quantum-mechanical theories of the solid state, is still widely applied today. The essence of the ideawas that a metal was a solid which contained free electrons that behaved rather like a gas and were conﬁned to a ‘box’ that was a representation of the shape of the bulk metal. In terms of this classical theory, the absorption of the light (electromagnetic) wave falling upon a metal induces an oscillation of the free electrons present. These, in turn, reradiate an electromagnetic wave which is recorded as scattering, as exempliﬁed by the Rayleigh and Mie theories described above. As with all oscillating 479 Colour in Metals, Semiconductors and Insulators systems, most light frequencies will only interact slightly with the electrons but, when the properties of the metal are appropriate, a particular frequency will be strongly absorbed, a phenomenon known as resonance. Pushing a swing illustrates this effect for a mechanical system. The frequency of the pushes will generally be out of synchronisation with the swing oscillations and not much energy will be transferred. However, when the frequency of the pushes just matches the normal oscillation frequency of the swing, a large amount of energy is transferred and the swing goes higher and higher. The frequency of the oscillation of the electron gas for a metal in air or a vacuum, which was found to be independent of the wave vector of the electromagnetic wave and the shape of the metal, was called the plasma frequency and is given by:

2 o2 ¼ ne p mee0 or, as n ¼ 2po,by

2 n2 ¼ ne p 2 4p mee0

where n is the density of free electrons in the metal, e is the charge on the electron, me is the electron mass and e0 is the permittivity of free space. Note that the electron mass in a nanoparticle, the effective mass, is usually different from the mass of a free electron in a vacuum. This equation suggests that a metal should be transparent for radiation with a frequency greater than the plasma frequency. In simple terms, the electrons cannot oscillate fast enough to interact with the electromagnetic ﬁeld. For radiation at frequencies less than the plasma frequency, this interaction is total, irrespective of wavelength. The electrons will absorb all of the incident radiation and then immediately reradiate it. Inserting values for the constants in the equation shows that the plasma frequency falls in the ultraviolet. Hence, metals should change from being transparent to being reﬂective in this radiation region, a prediction that is well obeyed. In reality, these classical collective electron oscillations are limited by quantum mechanical considerations. In this case, the oscillation is described as a plasmon, which is the quantum-mechanical particle corresponding to the collective oscillation of the bulk electrons in the metal. The absorption part, k, of the optical constants of the metal is now seen to arise from the plasmon oscillations in the metal. (Note, though, that the bulk optical constants of a metal may not apply to small particles.)

10.16.2 Surface plasmons and polaritons

From the point of view of metal nanoparticles, the most important plasmons are those at the surface of a metal, called surface plasmons. These can be imagined as electron density waves conﬁned, like ripples in water waves on the surface of a pond, to the metal surface. The collective oscillation of the electrons is called surface plasmon resonance. These may interact with incoming photons to form a hybridized quasiparticle called a surface plasmon polariton.9 Note that surface plasmon polaritons are often just called surface plasmons, although strictly speaking this is incorrect. The colour of metal nanoparticles, especially those of silver and gold, the most studied metal nanoparticles, is dominated by surface plasmon polariton formation. The peaks that appear in the extinction spectra of these

9 A polariton is hybridised state that forms when a photon couples with another excitation such as a phonon or a plasmon. Colour and the Optical Properties of Materials 480 small particles (Section 5.9) are due to the interaction of photons with surface plasmons. The recording of the peaks due to surface plasmons or surface plasmon polaritons is referred to as surface plasmon resonance spectroscopy. The detail of these surface waves is a function of the geometry of the surface and also of the optical constants of the surroundingp medium. For example, a planar metal film in a vacuum will support a surface plasmon wave offrequency op= 2, where op is the bulk plasma frequency. The plasmon so formed can move over the surface and is known as a propagating surface plasmon or propagating surface plasmon polariton. This is intriguing. It means that a photon, with a wavelength much greater than that of the foil, can be moved along the surface in this coupled way, opening the door to bypassing diffraction-limited optics (see below). When the oscillations are confined on the surface of a nanoparticle they are no longer free to travel, and so form localised surface plasmons or localised surfacep plasmon polaritons. For a spherical particle, the frequency of these confined oscillations is given by op= 3, where op is the bulk plasma frequency. Experimentally, gold spheres give strong absorption peaks in thewavelength range between 510 and 550 nm, and silver spheres in the range from 400 to 440 nm. Although not apparent from the simple theory just given, the absorption peak moves towards the red end of the visible as the sphere radius increases. In reality, the number and frequency of the surface plasmons on metal nanoparticles are dependent upon the shape and size of the particle. For gold and silver, these lie in the visible, and so contribute to the well-known colours displayed (Section 5.5). Much effort is now being focused upon the synthesis of nanoparticles of precise shapes, thus allowing tuning of the absorption. Cylindrical rods generally show two absorption peaks, one corresponding to the long dimension of the rod, the longitudinal surface plasmon polariton, and the other to the short dimension, the transverse surface plasmon polariton (Figure 10.43a). These produce two energy levels (Figure 10.43b) that give rise to two absorption maxima which, when added together, produce the observed colour of the particles. Because the energies are shape sensitive, the observed colour changes as the rod dimensions change. Shape tuning of the absorption characteristics of nanorods is limited, both by the geometry of the material and by the considerable synthetic skills needed to prepare the desired shape in sufficiently large quantities and purity. A second method of tuning the colour of the particles is to deposit shells of precious metal on an insulating core. In essence, two surface plasmon polaritons are generated, one on the outside of the metal sheath, corresponding to that for a solid spherical nanoparticle, and one on the inside of the metal sheath, corresponding to a surface plasmon polariton on the surface of a cavity in the metal (Figure 10.43b g). Because the metal shell is thin, these interact to give rise to two new energy levels similar to the formation of two molecular orbitals by the interaction of two atomic orbitals (Chapter 8). The process is called plasmon hybridisation. The energy levels of the new surface plasmon polaritons are given approximately by:

ho þ ¼ hos þ hoc

ho ¼ hoshoc

where os represents the plasmon contributed by the solid sphere component and oc represents the plasmon contributed by the cavity component (Figure 10.43h). The thicker the shell is, the weaker is this interaction and the closer the energies of the sphere and cavity plasmons become, until the two energy levels are the same and equal to that of a solid sphere. Only the lower energy level ho interacts strongly with the electric field of the light wave, but this is sufficient to provide a wide range of tunable frequencies, because the shell thickness, the core diameter and the total nanoparticle radius can be varied. Multiple shells can also be fabricated, giving yet more flexibility to the colour-varying abilities of these materials. As would be expected, the colours shown by these particles will change if they are embedded in materials other than air. 481 Colour in Metals, Semiconductors and Insulators

ω t transverse plasmon ω t 0

Energy ω longitudinal l ω (a) plasmon l (b)

plasmon on sphere 0 ω surface s Energy (c) (d) ω s

ω plasmon c on cavity 0 ω surface c Energy (e) (f)

ω +

ω c plasmons on shell 0 ω ω – + Energy ω s

ω (g) (h) –

Figure 10.43 Surface plasmon polaritons and associated energy levels for: (a), (b) metal nanorods; (c), (d) solid metal spheres; (e), (f) a cavity in a solid metal; (g), (h) a thin metallic shell on an insulating core. The energy scale is notional and the zero level is only to show relative positions

10.16.3 Polychromic glass

The relative contribution of scattering and absorption of small spherical metal particles is given by Mie theory (Sections 5.9 and 5.10). However, an explanation of the colours found required the introduction of the surface plasmon concept outlined above. An example of how the colours of small needle-shaped particles depend critically on particle dimensions is provided by a mid-twentieth century example of the fabrication of ‘polychromic’ coloured glass and highlights the careful processing that is necessary to achieve desired results. The colour-forming centres are minute silver needles. These form in glass after a complex set of heating cycles that initially results in the formation of sodium ﬂuoride cubes that form around tiny silver grains as nuclei. These cubes then act as nucleation sites for pyramids of a mixed sodium silver bromide phase, (Na,Ag) Br, that grow on the cubes of NaF, followed, ﬁnally, by the photochemical initiated growth of needle-like crystals of silver on the tips of the (Na,Ag)Br pyramids (Figure 10.44). The glass remains colourless when the crystals are below about 200 nm in size as they are too small to scatter light appreciably. If the crystallites become much larger than this then they scatter light and the glass becomes hazy or opalescent and has to be rejected. However, when the needles have dimensions of between 3 and 6 nm wide and between 3 and 36 nm in length then they are too small to cause much light scattering, but they do absorb strongly and generate brilliant colours when the glass is viewed in transmission. The precise absorption characteristics depend critically on the needle shape, especially the ratio of the width to the length (Table 10.4). In order to achieve Colour and the Optical Properties of Materials 482

NaF crystallite

(a)

(Na, Ag)Br pyramidal crystal

(b)

silver nanorod or elongated pyramid

(c) Figure 10.44 The formation of silver nanoparticles in polychromatic glass: (a) initial heat treatment forms cubic crystals of sodium ﬂuoride (NaF); (b) further processing causes silver halide crystals, mainly consisting of sodium bromide (NaBr), to grow on the cubic faces of the NaF crystallites; (c) needle tips become photoreduced to silver nanorods and pyramids a uniform bright colour the silver needles must all be of a similar size, a task needing considerable processing skill for success.

10.16.4 Photochromic glass

Photochromic glass is a material which is sensitive to light that owes its properties to silver nanoparticles. Although many types of photochromic glass have been fabricated, the best known are those which darken on exposure to high-intensity visible or ultraviolet light and regain their transparency when the light intensity falls. Such glasses arewidely used in sunglasses, sunroofs and for architecturalpurposes. (Forphotochromic plastics. see Section 8.13.) The mechanism of the darkening transformation is similar to that involved in the photosensitive glass described in the previous section. Photochromic glasses are complex materials which usually contain silver

Table 10.4 Colour and needle dimensions in polychromic glass

Needle length/nm Needle width/nm Length/width Colour transmitted 3.0 3.0 1.0 yellow 4.0 3.0 1.3 deep yellow 5.0 3.0 1.7 orange 6.0 3.0 2.0 red orange 7.5 3.0 2.5 red 10.0 3.5 2.9 magenta 12.0 3.5 3.4 purple 16.0 4.0 4.0 blue 21.0 4.5 4.7 turquoise 36.0 6.0 6.0 green 483 Colour in Metals, Semiconductors and Insulators

Melt Homogeneous Photochromic glass blank heat glass cast treatment

(a) (b) (c)

Cl Ag Cu

(d)

Figure 10.45 Photochromic glass: (a) glass melt containing CuCl and AgCl; (b) the melt is cast into a homogeneous glass blank; (c) heat treatment precipitates nanocrystals; (d) sodium chloride structure of AgCl containing copper impurities and Frenkel defects halides as the light-sensitive medium. The glass for this use would typically be an aluminoborosilicate (Pyrex type) material containing about 0.2 wt% of silver bromide or chloride. In addition, a small amount of a cuprous chloride (CuCl) is also added. When the glass is ﬁrst fabricated it is cooled rapidly. Under these conditions the silver and copper halides remain dissolved in the matrix and the glass produced is transparent and does not show any photochromic behaviour at all (Figure 10.45a and b). This material is transformed into the photochromic state by heating under carefully controlled conditions of temperature and time, which might be, for example, 550 C for 30 min followed by 650 C for 30 min. The heat treatment is chosen so that the halides crystallise in the glass matrix (Figure 10.45c). Care must be taken to ensure that the crystals do not become too large and that they do not aggregate. A desirable size is about 10 nm diameter and the individual crystallites should be about 100 nm apart. Contrary to the polychromic glass described above, in this case it is important that the processing conditions give a wide range of particle sizes, so that in effect the whole of the visible is uniformly absorbed. It is important that the copper is in the monovalent state and incorporated into the silver halide crystals as an impurity. Because the Cuþ has the same valence as the Agþ , some Cuþ will replace Agþ in the AgX crystal to form a dilute solid solution CuxAg1 xX (Figure 10.45d). The defects in this material are substitutional CuAg point defects. These crystallites are precipitated in the complete absence of light, after which a ﬁnished glass blank will look clear because the silver halide grains are so small that they do not scatter light appreciably. Light photons incident on the clear glass will liberate electrons from the Cuþ ions which are converted to 2þ . Cu ions (CuAg) in the process. These electrons are trapped by interstitial silver ions, which exist as Frenkel defects, to form neutral silver atoms:

. 0 hn þ CuAg ! CuAg þ e 0 þ . ! x e Agi Agi x þ x ! x Agi Agi 2Agi

This process continues until a small speck of silver is created. It is these clusters of silver which absorb the light falling on the glass. The absorption characteristics of the silver specks depend quite critically upon their size and shape. As mentioned, photochromic glass production is carefully controlled so as to produce a wide variety of shapes and sizes of the silver specks. For example, if the silver specks are rod shaped, each will have two Colour and the Optical Properties of Materials 484 absorption peaks, depending upon the ratio of length to width. Awide variety of shapes means that the whole of the visible spectrum is covered, ensuring that the glass darkens uniformly. In order for the glass to become clear again after irradiation it is essential that the silver particles release electrons to the Cu2þ ions when the light is turned off, reforming Cuþ ions and making the whole process reversible. This bleaching process is the reverse of the darkening process. In fact, the darkening and bleaching reactions are taking place simultaneously under normal circumstances, in dynamic equilibrium. When the amount of incident light is high, a large number of silver specks are present in the glass, hence leading to a high degree of darkening. At low light intensity the number of silver particles present decreases and the glass becomes clear again. Commercially useful materials require that the rate of the combined reaction is rapid. If the darkening takes place too slowly, or if the subsequent fading of the colour is too slow, the materials will not be useful. The presence of the copper halide is essential in ensuring that the kinetics of the reaction are appropriate and that the process is reversible.

10.16.5 Photographic ﬁlm

Photographic ﬁlm was the most widely used storage method for images throughout the twentieth century, and still has an important part to play in image capture and storage. Both black and-white and colour photography rely on nanoparticles of silver to capture images. The light-sensitive materials employed that give rise to the nanoparticles are silver halides, notably AgBr, which are dispersed in gelatine to form the photographic emulsion. Inorder toensure that the crystals are free ofmacroscopic defects such as dislocations,whichdegrade the perfection of the photographic images produced, the silver halide crystals are carefully grown within the gelatine matrix itself. The crystals so formed are usually thin triangular or hexagonal plates, varying between 0.01 and 10 mm in size, and in photographic parlance are known as grains. After illumination, some grains will have interacted with the light photons while some remain unchanged. Despite the fact that not all details of the photographic process are completely understood, the overall mechanism for the production of the silver particles is known and follows a path similar to that originally suggested by Gurney and Mott in 1938: 1. Interaction of a light photon with a halogen ion in the AgBr crystal. The energy from the photon hn liberates an electron from this ion:

hn þ Br ! e0 þ Br.

þ . 2. The liberated electron is free to move in the structure and migrates to an interstitial silver ion Ag (Agi ), x which is part of a Frenkel defect in the crystal, to form a neutral silver atom Ag (Agi ):

. þ 0 ! x Agi e Agi

3. In many instances, the above reaction will then take place in the reverse direction, and the silver atom will revert to the normal stable state as a Frenkel defect. However, the metal atom seems to be stabilised if another photon activates a nearby region of the crystal before the reverse reaction can take place, to produce a cluster of two neutral silver atoms:

x þ x ! x Agi Agi 2Agi

4. Further aggregation of Ag atoms occurs by a similar mechanism. 485 Colour in Metals, Semiconductors and Insulators

In this state the emulsion is said to contain a latent image. The ﬁlm is then put into a developer, which is a reducing agent. A grain that has interacted with light is totally reduced to metallic silver. The reactions taking place can be written down schematically as:

AgBr ðcrystalÞþlight photons !½AgBr crystal þ latent image ½AgBr crystal þ latent imageþdeveloper ! Ag crystal

All other crystallites remain unchanged. The ﬁnal step in the photographic process, ﬁxing, removes the unreacted silver bromide crystals from the emulsion, thus preventing further reaction (Figure 10.46).

silver halide crystal

(a)

light photons

(b)

crystal with latent image

(c)

silver crystal

(d)

(e)

Figure 10.46 Production of a negative image in a photographic emulsion: (a) ﬁlm emulsion; (b) interaction of some crystallites with light; (c) crystallites containing latent images; (d) development transforms crystallites containing latent images into silver grains; (e) ﬁxing removes all unreacted silver halide crystallites Colour and the Optical Properties of Materials 486

The image is stored in the emulsion by the silver crystallites. These are densely packed where the irradiance was high and are sparse where the irradiance was low. Bright areas on the image appear dark on the emulsion, and the result is a negative (image). A photograph (that is, a positive image) is created by exposing another emulsion layer, usually coated onto a sheet of paper, to light that has passed through the negative. One negative can produce as many positives (or prints) as needed. This simple picture ignores the fact that silver halides are not sensitive to the whole visible spectrum but respond mainly to short-wavelength (violet) light. This causes severe tonal problems in black-and-white photography, which relies upon a grey scale to indicate dark and bright parts of the image. Thus, on an untreated silver-halide-derived negative, blues, indigos and violets appear black and the other colours are only poorly registered. In a print (i.e. positive), the blues, violets and indigos appear to be far too pale. To broaden the sensitivity, the silverhalide crystals are treated with sensitizing dyes so that theyrespond to longer wavelengths. These dyes are adsorbed onto the surface of the silver halide crystals and absorb light energy, which is then transferred to the halide crystal, initiating the sequence of steps described above. The most widely used of these are derivatives of the cyanines (Section 8.6). The first dyes used extended the sensitivity into the yellow and green region; the result being the orthochromatic films. The overreaction to blue, violet and indigo could be corrected by using yellow filters, but because the film did not respond to red, the negative for red objects was clear and the resultant prints show all red areas far too dark. Later black-and-white films, so called panchromatic films, contained sensitizers that allowed all of the visible spectrum to be absorbed to an extent that the finished print showed colours in the expected tonal range, with blues indigos and violets appearing darkest and red appearing lightest. Colour films also relied upon the same silver halide processes. In this case the emulsion consisted of three layers, sensitive to blue, green and red. On processing, the exposed silver halides were replaced by dyes. Most colour films used the substractive primaries cyan, magenta and yellow. Processing led to either a positive (colour slides, for projection) or a negative (to be printed on paper) end result (see this Chapter’s Further Reading).

10.16.6 Metal nanoparticle sensors and SERS

Nanoparticle sensors, using colour change for the detection of chemicals, are readily available. Studies of this effect have centred upon the chemically inert precious metals gold and silver. One way in which colour change can be initiated is by formation of clusters. Clearly, if gold or silver nanoparticles cluster significantly then they are no longer quite as ‘nano’ and the observed colour will change. If the nanoparticles are treated with a surface layer that is sensitive towards an additive that can promote clustering, then the technique becomes an analytical one. For example, nanoparticles treated with DNA fragments can combine with complementary DNA fragments leading to a colour change that is detectable. Similarly, nanoparticles treated with surfactants so as to bind to a specific metal in solution can be used as an analytical test for the metal using colour changes of the nanoparticle suspension as the indicator. In this way, tests for toxic metals, such as arsenic in drinking water, have been developed. Apart from colour, metal nanoparticles have the unusual property of strongly enhancing the electromagnetic field close to their surface. When a photon interacts with a metal nanoparticle to create a surface plasmon polariton, the electromagnetic field is concentrated at the particle surface. In fact, the electromagnetic field can be enhanced by a factor of 104 or more. This enhanced field has a spatial range and contour that is dependent upon the shape of the nanoparticles. The effect has been used in a variety of ways, the best known being surface enhanced Raman spectroscopy (SERS). Raman spectroscopy is a well-established chemical analysis tool. It is based on inelastic scattering of photons from molecules. The majority of photons scattered by a molecule are scattered elastically and the behaviour is described by Rayleigh scattering theory (Sections 5.2 5.4). In essence, the molecule is treated like 487 Colour in Metals, Semiconductors and Insulators an antenna that simply reradiates the incident disturbance. However, molecules have rotational and vibrational energy levels, and in some circumstances a scattered photon can give up some of its energy to a molecule, exciting it to new vibrational or rotational levels. The inelastically scattered photon is then depleted in this small increment of energy, and has a lower frequency than the original. The output is called Stokes radiation. Similarly, if molecules are in excited vibrational and rotational levels then a scattered photon can pick up an increment of energy, allowing the molecule to drop down to lower energy levels. The departing inelastically scattered photon then has more energy and a higher frequency than the incident photons. This output is called anti-Stokes radiation. (The elastically scattered radiation, which suffers no energy change, is called Rayleigh radiation.) The change in energy of the inelastically scattered radiation is called the Raman effect. It occurs in about 1 in every 107 or so incident photons, and so the effect is very weak; nevertheless, monochromatic lasers with a high-intensity beam have allowed Raman spectroscopy to become an important tool in the study of molecular energy levels. This is because each molecule has a unique Raman spectrum that can be used as a fingerprint. The weakness of the Raman signal can be greatly increased by the surface electromagnetic field of a metallic nanoparticle. In effect, nanoparticles are coated with the molecules to be studied. The electric field of the incident photon is greatly enhanced at the metal surface, thus greatly increasing the Raman effect. The scattered photons are also similarly enhanced, so that the Raman signal is amplified by a factor of 106 108. This lies at the heart of SERS. To use the technique, nanoparticles of mainly gold (but silver and copper are also used) in a colloidal suspension or on a thin film are brought into contact with the material to be analysed. Molecules attach to the nanoparticles which are then examined by Raman spectroscopy. The amplification of the signal now means that even single molecule attachment can be detected. However, the surface enhancement of the electromagnetic field is sensitiveto nanoparticle geometry, and carefully prepared colloids with a uniform shape and narrow size distribution is essential for the work.

10.17 Extraordinary Light Transmission and Plasmonic Crystals

In 1989 Ebbesen discovered that a thin gold film perforated by small holes, of diameter much less than the wavelength of visible light, deposited onto a glass slide, was able to transmit light very well, although, simplistically, no light should be transmitted at all. The phenomenon, called extraordinary light transmission, was fairly complicated, in that although some wavelengths of light were transmitted with an unusually high intensity, other wavelengths were not transmitted as well, so that objects viewed through the foil were coloured. The explanation of the effect, which took 10 years to unravel, was that the incoming photons interacted with surface plasmons at the metal dielectric interface, were transported through the holes in the foil and were then reradiated. The colours transmitted are those near to the natural oscillation frequency of the plasmons; that is, the transmission spectrum of the film shows peaks at frequencies corresponding to the excited surface plasmon modes. These, however, depend upon the geometry of the array of holes and their sizes. The consequence is that the optical transmission of the perforated foil can be changed by adjusting the geometry and disposition of the holes. In the simplest cases, holes, circular or square, are arranged on a crystal-like lattice, with a repetition defined by a ‘unit cell’. Other surface geometries have also been explored, including regular arrays of nanopyramids. As well as the hole geometry, the surrounding medium is also important. If this is different on each side of the film, as when a glass slide is used as a substrate, and the whole is viewed in air, the surface plasmons formed on each side of the film have different frequencies. This means that the transmission spectra consist of two sets of peaks, offset by the difference in the refractive indices of the insulating medium in contact with the metal film. Colour and the Optical Properties of Materials 488

Metal ﬁlms perforated or patterned in a regular array, by dimples, holes slits or grooves and so on, separated on a nanometre scale are called plasmonic crystals. The optical behaviour of these objects is dependent upon the nature of the metal, the nature of the patterns and the interface between the metallic and insulating surrounding medium. This latter property allows the device to be used as a sensor for molecules or molecular layers deposited on the surface. In this way it has been possible to detect minute quantities of absorbed material and to differentiate between closely related molecular species.

Further Reading

An introduction to band theory adequate for this book is R. J. D. Tilley, Understanding Solids, John Wiley and Sons, Ltd, Chichester, 2004. Much information on colour centres and the colours of irradiated minerals is given by K. Nassau, Gems Gemol. XIV, 343 355 (1980). K. Nassau, The Physics and Chemistry of Color, 2nd edition, Wiley-Interscience, New York, 2001. Dental phosphors and related materials are described by J.-M. Spaeth, Radiat. Meas. 33, 527 532 (2001). For a complete discussion of solid-state lighting, especially with reference to LEDs, see C. J. Humphreys, Mater. Res. Soc. Bull. 33, 459 470 (2008). LEDs and diode lasers: S. Nakamura, Mater. Res. Soc. Bull. 22 (February), 29 35 (1997). S. Nakamura, Mater. Res. Soc. Bull. 23 (May), 37 43 (1998). N. Holonyak Jr, Mater. Res. Soc. Bull. 30, 509 517 (2005). M. Fox, Optical Properties of Solids, Oxford University Press, 2001, Chapter 5, esp. 5.4. B. E. A. Saleh and M. C. Teich, Fundamental of Photonics, John Wiley and Sons, Inc., New York, 1991, Chapters 15 and 16. OLEDs: M. Thompson, Mater. Res. Soc. Bull. 32, 694 701 (2007). S. So, J. Kido, P. Burrows, Mater. Res. Soc. Bull. 33, 663 669 (2008). S. Ye, Y. Liu, C.-A. Di, H. Xi et al., Chem. Mater. 21, 1333 1342 (2009). Dendrimers in OLEDs: J. Li, D. Liu, J. Mater. Chem. 19, 7584 7591 (2009). Electrochromic displays, including ‘smart’ windows are described by P. Monk, R. Mortimer, D. Roseinsky, Electrochromism and Electrochromic Devices, Cambridge University Press, 2007. A. A. Argun, P.-H. Aubert, B. C. Thompson, I. Schwendeman et al., Chem. Mater. 16, 4401 4412 (2004). R. Baetens, B. P. Jelle, A. Gustavsen, Sol. Energ. Mater. Sol. Cells 94, 87 105 (2010). The PANI PSS/PEDOT PSS electrochromic device that served as the basis for text discussion is detailed in L.-M. Huang, C.-H. Cheng, T.-C. Wen, Electrochim. Acta 51, 5858 5863 (2006). The historical evolution of solar cells can be followed via the following sources: Y. Hamakawa, Sci. Am. 256 (April), 77 82 (1987). 489 Colour in Metals, Semiconductors and Insulators

Various authors, Mater. Res. Soc. Bull. 18, (October), 18 66 (1993). Various authors, Mater. Res. Soc. Bull. 30, 10 52 (2005). Various authors, Mater. Res. Soc. Bull. 32, 211 247 (2007). D. Ginley, M. R. Green, R. Collins, Mater. Res. Soc. Bull. 33, 355 364 (2008). The historical evolution of dye-sensitized solar cells can be followed via the following sources: B. O. Regan, Nature 353, 737 740 (1991). M. Gr€atzel, Mater. Res. Soc. Bull. 18, (October), 61 66 (1993). M. Gr€atzel, J. Photochem. Photobiol. C Photochem. Rev. 4, 145 153 (2003). M. Gr€atzel, Mater. Res. Soc. Bull. 30, 23 27 (2005). S. Ahmad, J.-H. Yum, Z. Xianxi, M. Gr€atzel, H.-J. Butt, K. Nazeerruddin, J. Mater. Chem. 20, 1654 1658 (2010) and references cited therein. Photodiodes and charge coupled devices are described in: B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, John Wiley and Sons, Inc., New York, 1991, Chapter 17. Much technical information concerning digital photography can be obtained from the websites of microscope and camera manufacturers, including Nikon, Olympus, Canon and so on. Polychromic glass is described in D. M. Trotter, Sci. Am. 264 (April), 56 61 (1991). S. D. Stookey, G. H. Beall, J. E. Pierson, J. Appl. Phys. 49, 5114 5123 (1978). For a concise introduction to electromagnetic waves in solids and the derivation of the plasma frequency, see L. Solymar, D. Walsh, Electrical Properties of Materials, 7th edition, Oxford University Press, Oxford, 2004, Chapter 1. More detail, clearly presented, is in N. Braithwaite (ed.), Electromagnetism, Book 3, Electromagnetic Waves, The Open University, Milton Keynes, 2006. A survey of the surface plasmonic properties of metallic nanoparticles is given in a series of articles by various authors in Mater. Res. Soc. Bull. 30, 338 389 (2005). The photographic process is described, together with many references, in the following reviews and in much technical literature produced by the manufacturers of ﬁlm: F. C. Brown, The photographic process, in Treatise on Solid State Chemistry, Vol. 4, Reactivity of Solids, N. B. Hannay (ed.), Plenum, New York, 1976, Chapter 4. J. A. Kapecki, J. Rodgers, Colour photography, in Encyclopedia of Imaging Science and Technology, Vol.1, J. P. Hornak (ed.), John Wiley and Sons, Inc., New York, 2002. S. H. Mervis, V. K. Walworth, Instant photography, in Encyclopedia of Imaging Science and Technology, Vol. 1, J. P. Hornak (ed.), John Wiley and Sons, Inc., New York, 2002. Plasmonics and plasmonic crystals are described by H. A. Atwater, Sci. Am. 296 (April), 38 45 (2007). Various authors, Mater. Res. Soc. Bull. 30, 338 380 (2005). J. Heber, Nature 461, 720 722 (2009). T. W. Odom, Mater. Res. Soc. Bull. 35, 66 73 (2010). The ﬁrst report of extraordinary transmission was T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, Nature 391, 667 669 (1998).

Index

Note: Figures and Tables are indicated (in this index) by italic page numbers, footnote by sufﬁx ‘n’

Abbe number, 67 Airy rings, 202, 203, 226 Abbe numerical aperture value, 204 Airy wavefront, 205 Abbe V-value, 67 alanine, 165 abscission layer, 332 albite, 366 absorbance, 36 Alexander’s dark band, 69, 71,75 absorption, 17, 18, 33 4 alexandrite, 283 4, 286 double, 297 algae, chlorophylls in, 320 Einstein coefficient, 20 alizarin, 352 rate of, 18 21 alkali metal halides, F centres, 430, 431 absorption band, 65, 66 alkali metals, (energy) term for, 252 absorption coefficient, linear (Napierian), 35 alkaline earth compounds, colours, 255, 315 absorption edge, 290, 420 alumina, 43, 189 absorption efficiency (factor), 186 aluminium, 478 absorption index/coefficient, 51, 477 aluminium nitride, 441 absorption spectrum aluminium oxide: see alumina; corundum Nd3þ -doped glass, 289 aluminium oxynitride, 43 and polarized light, 151, 279 80 amethyst, 431, 432 rhodamine 6G dye, 356 amino acids, enantiomers, 165, 166 solar radiation, 255, 256 ammonites, 140 transition metal ions, 264, 265, 275, 278 ammonium dihydrogen phosphate (ADP), 153 water and ice, 315 16 ammonium iron(III) citrate, 342, 343 acceptor dopants, 424 amplification, 18, 19,22 acceptor (in quenching), 376 amplitude diffraction grating, 198, 206, 207 achromats, 68 amplitude grating, 198 ACTFEL (AC thin-film electroluminescent) display, 391, 395–6 amplitude hologram, 241 actinide, 301n amplitude object, 198 actinoid compounds, colours, 295 amplitude of wave, 7, 44 actinoids, 249 analyser, polarization, 137 activators, 366, 379 anatase, 121, 474 active matrix display, 39, 170 angular frequency, 44 additive colour mixing, 29 31 anhydrobase, 330 on displays, 170, 447, 463 aniline purple, 337 ADP (ammonium dihydrogen phosphate), 153 anisotropic materials, 138 aequorin, 416 anodically coloured materials, 463, 467, 468, 469 aether drift, 2 anodized films, 103 4 afterglow, 373 4, 387 anthocyanidins, 324 6, 329 agate, 366 anthocyanins, 323, 324, 325, 329, 332 aggregation, 435 anthracene, 378 aglycons, 324, 329 antibonding molecular orbital, 310, 457 AGS (silver gallium sulfide), 153 antimony ions, in phosphors, 379 80 AGSe (silver gallium selenide), 153 antireflection (AR) coatings, 62, 64, 105 10 air lens, 64 graded-index, 108 10 airlight, 180, 181 moth-eye, 64, 108, 109, 231 2 Airy disc, 202, 203 in solar cell, 471, 472

Colour and the Optical Properties of Materials Richard J. D. Tilley Ó 2011 John Wiley & Sons, Ltd Index 492 antireflection (AR) layer, 106 7, 109 Beer Lambert Bouguer law, 35 tuneable, 107 Beer Lambert law, 35, 36, 176 anti-Stokes fluorescence, 394 Beer’s law, 34, 176 anti-Stokes radiation, 487 beetles, colours, 223, 230 appearance of objects, 40 3 bending mode, water molecules, 315 apple, colours, 328 Benton holographs, 239 aquamarine, 344 5 benzopyran, 359 aragonite, 366 Berlin blue, 342 Archer fish, 49 beryl structure, 283, 286, 344 5 argon-ion laser, 115 biaxial crystal arsenic compounds, colours, 297 double refraction in, 144 7 arsenic poisoning, 297 trichroism in, 149 arsenic triselenide glass, 80 bimolecular reaction, 378 arsenic trisulfide, 437 biofluorescence, 414 astaxanthin, 319 bioluminescence, 366, 414 16 astronomical telescopes, 204, 478 biophosphorescence, 414 atomic absorption analysis, 255 birefringence, 138, 141, 147 atomic orbitals, 264, 266, 267, 300 circular, 167 attenuation, 34, 176 colour produced by, 147 extrinsic, 79 phase matching and, 160 factors affecting, 176 stress, 148 in optical fibres, 79, 80 bis(dimethylglyoximate)nickel(II), 350, 351 attenuation coefficient, 34, 36, 186, 477 bismuth borate, 155 attenuation cross-section, 36 black vacuum, 257 aurora australis, 313 black-and-white photographic film, 486 aurora borealis, 313 black-and-white television, 387 9 autofluorescence, 367 black-body radiation, 11 13 autumn leaf colours, 332, 333–4, 335 law, 11 13, 15, 20, 44, 45, 119 auxochrome, 317 bleaching auxochromic shift, 324 electrochromic film and materials, 463, 467, 468 Avogadro’s number, estimation of, 178 eye pigments, 24, 26 axis photochromic materials, 360, 484 fast, 138 blinking, of core shell quantum dots, 456, 458 optic, 139, 140 Blu-Ray discs, 94, 448 slow, 138 blue butterflies, 122, 123–4, 188, 324, 326 azolitmin, 351 blue diamonds, 428 azurite, 284, 285 blue eyes, 188 blue feathers, 188, 234 5 b-radiation, measurement of, 417 Blue John, 430 1 Babinet’s principle, 200, 201, 205 blue moon, 180, 188 backscattering efficiency, 194, 195 ‘blue remembered hills’, 179 80, 181 bacteriorhodopsin, 28 blue shift, 374, 451 Balmain’s paint, 365 blue sky, 23, 40, 178 9, 180 Balmer series, 250, 251 blue sun, 188 band blueprints, 342 4 conduction, 4, 5, 388, 419, 420 Boltzmann constant, 17, 43 energy, 419 Boltzmann’s equation (for specific rotation dispersion), 167 valence, 4, 5, 388, 419, 420 Boltzmann’s law, 17 18, 20 band edge, 420 bonding molecular orbital, 310, 457 band gap, 4, 419, 420, 421 boron-doped diamonds, 428, 429 direct, 442, 443 Bouguer’s law, 34, 176 indirect, 442, 443 see also Beer Lambert Bouguer law listed for various oxides, 422 bowing coefficient/parameter, 441 optical, 420, 422 Brackett series, 251 band structure, 4, 5, 420, 421 Bragg angle, 213 band theory, 419 Bragg equation, 211 13 bandgap engineering, 454 modified (for opal), 215 17 bandpass filters, 114 Bragg fibre grating, 115 19 barite, 366 Bragg reflector, 114 15, 117 barium chromate, 346 Bragg’s law, 211 13 barium compounds, colours, 255, 315 applications, 223, 229, 230 barium fluorobromide, 436 dynamical form, 224 5 barium magnesium aluminate, 385 see also Bragg equation barium titanate, 391 Brewster angle, 134, 135 bathochromes, 26 Brewster’s law, 133 4 bathochromic shift, 26, 317, 374 Brewster window, 134, 135 Bayer filter, 476 brightening agents, 368 BBO (beta barium borate), 153, 157 brightness, 28 beam splitter, 56,57 see also illuminance 493 Index bromocresol blue, 352 charge-transfer processes, 340 bromophenol blue, 352 charge-transfer transition bromothymol blue, 352, 353 anion-to-cation, 345 6 butterfly cation-to-cation, 340, 341 colours, 122, 123–4, 324, 326, 340 cation-to-ligand, 340, 374 eye, 108, 109 intervalence, 340, 341 wing markings, 114, 115 intra-anion, 348 9 wing scales, 29 30, 122, 123–4, 188 ligand-to-cation, 340 ligand-to-ligand, 340, 374 cadmium orange, 441 chemical analysis, 254 5 cadmium selenide, 441, 455 chemiluminescence, 366, 413 14 ZnS-coated (quantum dot), 350, 351, 456 china clay, 188 cadmium sulfide, 437, 441 Chinese blue, 342 photovoltaic cell, 471, 472 chiral carbon atom, 164, 165, 166 quantum dot, 455, 457 chiral centre, 164, 165, 166 cadmium telluride, 472 chiral molecule, 164 cadmium yellow, 437 chiral nematic (liquid crystal) phase, 228, 229 calcite chlorophyll, 320, 321–2, 332 double refraction of, 139 40, 140, 142 cessation of production, 332 fluorescence, 366 chloroplasts, 332 structure, 139, 141, 142, 143 cholesteric blue phases, 230 calcium carbonate, 139 cholesteric liquid-crystal mesophase, 168, 228 30 minerals, 188, 366 cholesterol-based compounds, 168 calcium chromium silicate, 295 chroma, 28 calcium fluorophosphate, 379 chromatic aberration, 68 calcium sulfide, 365 chromaticity diagrams, 30 1, 32–3 calcium tungstate, 292 chrome alum, 284, 286 candle flame, 11, 14, 314 chrome green, 277, 295 carbon bonds chrome yellow, 346 conjugated double, 317 chromic oxide, 277, 295 double, 317 chromium compounds, colours, 286, 295 carbon dioxide (CO2) laser, 156 chromogen, 317 carbon nanotubes and nanorods, 136, 450 chromophore, 316 17 carborundum, 103 chrysoberyl, 283, 284, 286 a-carotene, 317, 318 C.I. fluorescent brightening agent 30, 368 b-carotene, 317, 318 C.I. Solvent Yellow 124, 354 5, 356 carotenoids, 317 19, 332 CIE 1931 chromaticity diagrams, 31, 32–3 carrot, colour, 317 cinnabar, 437, 438 cathode rays, 385 6 circular birefringence, 167 cathode-ray tube (CRT), 365, 386 circular dichroism, 167 television tube, 386 9 circularly polarized light, 130, 131 cathodically coloured materials, 464, 467, 469 citrine, 431 cathodoluminescence, 365, 366, 385 90 Clebsch Gordon rule, 304, 305 cationic configurations, 301 close-packing of spheres and atoms, 218, 219 Cauchy’s equation, 65 6 CMK (cyan/magenta/yellow) colour model, 37 CCD: see charge couple device CMYK colour model, 39 CD: see compact disc co-activator, 379 ceramics, 188, 189 90 cobalt aluminate, 295 6 pigments for, 295 7 cobalt chromite, 296 7 transparent, 43, 189 90 cobalt compounds ceria (CeO2), 51, 420, 421 colours, 285, 286, 295 7 cerium compounds in glass, 285, 287, 288 absorbance spectrum, 291 cobalt silicate, 296 in sunscreens, 346 coelenterazine, 415 cerium ions coherence length, 160 colours, 289, 290 coherent light, 7, 8, 17, 235 energy levels, 291, 383, 393 coherent scattering, 197 in phosphors, 393 4 cold light, 363 cerium oxide, europium-doped, 397–8 collagen, 162, 163, 233, 234 chalcone, 323, 325 of inorganic molecules, 311 15 chalcopyrite, 437, 439 colloidal crystals, 218, 220, 230 charge carriers, strongly confined, 450 colloids, 478 charge couple device (CCD), 474 6 coloration ‘dark’ current, 476 additive, 29 31, 170, 447, 463 4 dynamic range, 476 subtractive, 37 9, 194 full-well capacity, 476 colorimetric sensor films and arrays, 353 4 photography using, 476 7 colour spectral response, 476 7 and birefringence, 147 charge-transfer colours, 340, 341 2, 344 9, 346, 374 of butterflies, 122, 123–4, 324, 326, 340 Index 494 colour (Continued) HVC, 28 charge-transfer: see main entry: charge-transfer colours RGB, 30 complementary, 31, 38 colour photographic film, 486 of copper compounds, 264, 265, 275, 284 5, 286 colour printers, 37, 39 crystal-field, 264 70, 284 6 colour rendition, sodium-vapour lamps, 263 and diffraction, 198 203, 205 8 colour spaces, 28 of electrochromic polymers, 468, 469, 470 colour television, 170, 384 5, 388, 389 and energy, 3 4 colour temperature, 13 of eye, 188 correlated, 13 14 of flowers, 323 8, 329 of incandescent objects, 14 of fluorescent proteins, 407 colour triangle, 30 1 gamut of, 31 combinatorial tones, in water molecules, 315, 316 of gemstones, 150 1, 214 15, 277, 283 4, 285, 286, Commission Internationale de l’E´ clairage, 31 344 5 see also CIE of incandescent objects, 11, 13 14 Common Blue butterfly, 122, 123–4, 324, 326 of insulators, 420 compact discs (CDs), 57, 94, 209 10, 448 iridescent, 91, 94, 122, 230 laser used, 448 of lanthanoid ions, 288 90 recordable (CD-R), 94 of leaves, 321–2, 332, 333–4 reflection grating colours from, 209 10 meaning of term, 1, 3 rewritable (CD-RW), 94 of metallic nanoparticles, 478 87 complementary colours, 31, 38 of metals, 477 8 in thin films, 100, 103, 126–7 of minerals, 122, 124, 125, 284 5, 286 complex numbers, 51n mixing of, 29 31, 37 9, 170, 447, 463 4 complex refractive index, 51, 191, 477 of nickel compounds, 264, 265, 274, 277, 278, 286 complexes, 350 perception of, 10, 23, 28, 29,45 computer displays, 170, 173 of pigments, 295 7, 322, 333 40, 437 computer memory, 474n of polychromic glass, 482 concentration quenching, 377 8 primary, 30, 37 conduction band, 4, 5, 388, 419, 420 quantum dot, 455, 456 cone opsins, 24 of red wine, 328 32 configuration interaction energy, 276, 277 and reflection, 91 128 conjugated double bonds, 317, 458 and refraction, 67 75 in cyclic compounds, 319 23 of ruby, 150, 277 81, 286 constructive interference, 8, 96, 97, 98, 228, 235 saturated, 31 continuous spectrum, 11 of semiconductor alloys, 441 conversion factors, 43 of semiconductors, 436 9 cooperative luminescence, 394 of shells, 122 copper as structural probe, 287 colour, 478 suppression of, 120 reflectivity, 477 of thin films, 99 104, 126–7 copper acetoarsenate, 297 of transition metal ions and compounds, 264 70, copper compounds 284 5, 286 absorption spectrum, 264, 266, 275 of water, 315 16 colours, 264, 265, 275, 284 5, 286 colour blindness, 24, 31 flame colour, 255 colour centre, 429 36 Orgel diagram, 275 complex, 433, 434 5 copper indium selenide, 472 electron-excess, 430 1 copper ions, detection of, 350, 351 hole-excess, 430, 432 3 copper nanoparticles, 487 surface, 434 copper phthalocyanine, 322, 323 see also F centre copper selenides, mixed, 472 colour centre laser, 434 5 copper sulfide, 437 colour change corals, colours, 407 auxochromic, 324 cordierite, 184 bathochromic, 26, 317, 374 core shell composites/nanodots, 412, 456, 458 hyperchromic, 317 cornea (of eye), 190, 198, 233 4 hypochromic, 317 cornflower, colour, 324 hypsochromic, 317, 374 corona, 226 colour-change sensors, 349 55 corpuscular theory of light, 1, 2, 3 colour confusion, 31 correlated colour temperature, 13 14 loci of, 31, 33 corundum (Al2O3) colour filters, 37, 38 Cr3þ in, 277, 286 colour models, 28 doping of, 284, 345 CMY, 37, 486 fluorescence, 366 CMYK, 39 refractive index, 61, 66 HCL, 28 Ti3þ in, 282, 283 HIS, 28 Ti4þ and Fe2þ in, 345 HSB, 28, 29 cosmic microwave background radiation, 13 HSL, 28 counter electrode (in solar cell), 473 495 Index coupling, 252 degenerate semiconductors, 440 see also j j coupling; Russell Saunders coupling; spin orbit 3-dehydro-retinal (retinal2), 26 coupling delafossite-structure oxides, 440 crests and troughs, 6,7 delphinium, colours, 329 critical angle, 54, 55, 134, 135, 445 dendrimers, 462 crocetin, 318, 319 dendrites, 462 crocin, 318, 319 density, and refractive index, 60 2, 138 cross-relaxation, 369, 377, 399, 401, 403 destructive interference, 8, 9, 96, 99, 235 crustacean detergents, fluorescent brightener in, 368 blood, 322 3 dextrorotatory molecules, 164, 165, 166 colour, 319 DFG (difference frequency generation), 155 a-crustacyanin, 319 DFM (difference frequency mixing), 155 crystal diamond anorthic (triclinic), 137 band gap in, 424 5, 437 biaxial, 144 7, 149 blue, 428 birefringence of, 138 boron impurities in, 428, 429 centrosymmetric, 155 ‘canaries’, 425 colloidal, 218, 220 Cape yellow, 425 cubic (isometric), 138, 143 impurity colours in, 425 8, 446 dichroic, 149, 150 N V centre, 428 hexagonal, 138, 143, 165 N2 centre, 427 8 liquid: see main entry: liquid crystal N3 centre, 427 metamaterial, 85, 86 nitrogen impurities, 425 8 monoclinic, 138, 144 spectral colours (‘fire’), 68 non-centrosymmetric, 137, 157 structure, 424, 426 nonlinear, 151 3, 154 5 thin films, 428 optically negative, 143, 147 yellow, 425 optically positive, 143, 146 dichroic glass, 193 orthorhombic, 138, 144, 283 dichroic sheet polarizer, 136 photonic, 85, 218, 220, 223 dichroism, 149, 150 pleochroic, 149, 151 circular, 167 tetragonal, 138, 143 in gemstones, 150 1, 279 81, 283 trichroic, 149 dichromated gelatine, 242 triclinic, 138, 144 dichromats, 24 trigonal (rhombohedral), 138, 140, 143 dielectric constant, 58 uniaxial, 143 4, 149 dielectric mirrors, 111 12 unit cell of, 138 dielectric susceptibility, 152 crystal defects, 445 diffraction, 33 crystal field by amorphous material, 225 6, 227 intermediate, 273 7 Bragg’s law, 211 13 octahedral, 271, 272, 274 6 by a circular aperture, 202 3 splitting, 266, 268, 269 70 colour production by, 198 203, 205 8 strong, 270 1 by crystals, 211 25 tetrahedral, 271, 272 3 by disordered gratings, 225 31 weak, 271 3 by droplets, 226 7 crystal-field colours, 264 70, 284 6, 345, 346 by dust, 226 7 crystal symmetry dynamical theory, 213, 224 5 and refractive index, 137 9 of electrons: see main entry: electron in ruby crystal, 279 80 diffraction crystal systems, 137 Fraunhofer, 198 cyanidin, 325, 329 Fresnel, 198 cyanin, 326, 327, 329 from disordered gratings, 225 31 cyanotype process, 342 4 images limited by, 87 cyclamen, 321 kinematical theory, 213 by moth-eye structures, 231 3 d-orbitals, 264, 266, 300 by opal, 213 18 crystal-field splitting, 266, 268 by a rectangular aperture, 200 1 degeneracy of, 267, 269 by a slit, 198 200 shapes, 266, 267 by specks, 226 7 Daltonism, 24n by sub-wavelength structures, 231 5 dangling bonds, in quantum dots, 455 wavelength estimation by, 210 11 dark adaptation, 28 of X-rays: see main entry: X-ray diffraction data storage, 94, 297, 474n diffraction grating, 198 daylight white, 31 colour production by, 205 8 decorative coatings, 210 disordered, 225 31 defects: see crystal defects; Frenkel defects; linear, 205 8 point defects moth-eye surface as, 109, 232 3 defoliation, mechanical, 104 see also grating(s) degeneracy of orbitals, 267, 269, 271 diffraction grating equation, 205 Index 496 diffraction limit, 203 5 effective refractive index, inverse opals, 221 3 Abbe criterion, 204 effective temperatures (of stars), 14 Rayleigh criterion, 203, 204 egg yolk, colour, 319 diffraction pattern Egyptian blue, 297 from amorphous material, 225 6 Einstein coefficient Fraunhofer, 198, 199 for absorption of radiation, 20 orders in, 198, 205 for spontaneous emission, 19 from random droplets or specks, 226 7 elastic scattering, 33, 175 spectra, 200 elbaite, 149 Whewell Quetalet, 227 electric dipole transition, 254 digital camera display screen, 170, 172 electric field vector, 5, 6, 129 digital photography, 170, 172, 474 7 electrochromic device, 463, 464 dimethyl glyoxime, 350 asymmetric arrangement, 463, 464 diode: see light-emitting diode dual arrangement, 463, 464 diode laser, 448 with tungsten trioxide film, 465 6 diopside, 366 electrochromic film, 463, 464 70 1,2-dioxetanedione, 413, 414 bleaching of, 464 diphenylamine, 354, 355 electrochromic materials 9,10-diphenylanthracene, 414, 415 anodically coloured, 463, 467, 468, 469 direction bleached, 464 allowed, 135 cathodically coloured, 464, 467, 469 fast, 138 inorganic, 465 8 slow, 138, 141 organic, 468 vibration, 135, 140, 141 polymeric, 468 70 dispersion, 65 electrochromic reactions, 464, 466, 467, 468 anomalous, 65, 66 electroluminescence, 366 intermodal, 82 molecular, 457 9 modal, 82 organic, 457 9, 460 normal, 65, 66 electroluminescent displays, 391 4, 446 in optical fibres, 81 2, 83 electromagnetic spectrum, 2 production of colour by, 67 75 electron specific rotation, 167 confined, 450 and wavelength, 82 effective mass, 451, 479 dispersive power, 67 energy in quantum structures, 451 display(s) subbands (energy levels), 451, 453, 454 electroluminescent, 391 4 electron configurations field-emission/field-effect, 390 1 lanthanoids, 301, 303 interference-modulated, 110 11 lighter atoms, 300 1 liquid-crystal, 169 73 listed for various atoms, 249, 300 1, 302 plasma, 259, 383 5 mercury, 263 thin-film electroluminescent (TFEL), 391 4 neon, 261 distributed Bragg reflectors, 114 15 sodium, 263 DNA molecules, response to stretching, 410 transition metals, 301, 302 donor dopants, 424 electron diffraction, 213 donor (in quenching), 376 dynamical theory, 213 donor p-bridge acceptor molecules, 406 electron diffraction pattern, 214 Doppler effect, 248 electron electron repulsion, 252, 253, 254, 266, double absorption, 297 271, 302 double refraction, 139 electron-excess centres, 430 1 in biaxial crystal, 144 7 electron gun, 386 of calcite, 139 40, 142 electron hole, 421 in uniaxial crystal, 143 4 electron hole pair(s), 423, 456, 471, 472, 475, 476 doublet states, 303 electron microscopy, 87, 390 drift, 472 electronic energy levels, 310, 311 Drude Lorentz free electron theory, 478 electronic ‘paper’, 39 40 drying agents, 285 electronic transitions, 258, 262, 263 4 DsRed fluorescent protein, 407 absorption due to, 79 DVDs (digital versatile/video discs), 57, 94, 210, 448 electrophoresis, 39, 40 DWDM (dense wavelength division multiplexing), 119 elliptically polarized light, 130, 131 dye lasers, 355 8 embossed holograms, 242 3 dye-sensitized solar cells, 472 4 emerald, 283, 286 dyes, 322, 333 40 emeraldine, 468, 469 in glow sticks, 414 base form, 469 in solar cells, 473 4 salt forms, 469 dynamic quenching, 374, 378 emission Einstein coefficient, 19 e-books, 39 40 rate of, 18 21 e-ink process, 39, 40 spontaneous, 17, 18,19 e-ray (extraordinary ray), 139, 141, 142 stimulated, 17, 18, 19, 21, 22, 259, 260, 282, 292, 369 497 Index emission spectrum compound eye, 108, 109, 231 phosphors, 380, 382, 394 dark adaptation, 28 rhodamine 6G dye, 356 diseases, 162 sodium vapour lamp, 263 insect eye, 108, 231 2 tuning in quantum structures, 454 mirror eye, 125 6 enantiomers, 164, 165 6 moth eye, 64, 108, 109, 232 energy photoreceptors: see main entry: photoreceptors absorption and emission, 368 70 scallop eye, 125 units, 43 sensitivity, 24, 25 energy bands, 419 structure, 24, 190 energy exchange efficiency, 376, 377 energy exchange equation, 309, 310 F centre(s), 429 30 energy level(s), 4, 253 listed for various alkali metal halides, 430 deep, 424 Fabry Perot etalon, 110 in intermediate crystal field, 273 7 face-centred cubic structure, 218 of many-electron atom, 306 7 Fairy primrose, 323 molecular, 309 11 feathers, 188, 234 5 shallow, 424 feldspars, 122, 124 in strong crystal field, 270 1 Fermi energy, 460 in weak crystal field, 271 3 Fermi level, 459, 460, 474 energy-level diagrams ferric oxyhydroxides, 346, 347 inert gases, 258 ferroelectric crystals, 242 lanthanoid ions, 291, 292, 293, 381, 383, 390, 393, 447 ferroprussiate paper, 342 lasers, 261, 293 fibre Bragg gratings (FBGs), 115 19 mercury atoms, 264 fibre optics, 75, 77 84 molecular fluorophore, 406 see also optical fibres OLED, 462 field-emission display (FED), 390 1 sodium atoms, 262 filling factor, 108 energy-level populations, 17 18 film energy transfer (in quenching), 369, 377 8, 399 401 anodized, 103 4 erbium-doped optical-fibre amplifiers, 294 5, 446 birefringent, 151 erbium ions electrochromic: see main entry: electrochromic film colours, 289 photographic: see main entry: photographic film energy levels, 447 polymer, 148, 149, 151 erythrolitmin, 351 see also thin film 2-ethylanthraquinone, 354, 356 filter eumelanin, 337, 340 bandpass, 114, 116 europium ions interference, 114 cerium oxide doped with, 397–8 longpass, 114, 116 colours, 289, 290 optical, 38, 114, 224 energy levels, 292, 381, 390, 393, 447 polarization, 136, 137 in phosphors, 381, 382, 385, 389, 392 3, 436 shortpass, 114, 116 evanescent waves, 54, 56, 57, 87, 89 firefly, 415 excited state, 17 fireworks, colours, 255, 315 excited-state absorption (of photons), 369, 370, 396 7, 399, 401 first-order kinetics, fluorescence and phosphorescence, 372, 373 exciton, 421 4, 458 fishnet structures, 85, 86 free (Mott Wannier), 423 flame colours, 255, 255, 315 in molecular crystals, 424 flame test, 255 singlet, 458, 459, 460 flashlamps, 257 tightly bound (Frenkel), 423 4 flashtubes, 257 triplet, 458, 459, 460 flat band model, 419, 420, 421 exciton blocking layer, 461, 462 flavone(s), 323, 325 exciton energy levels, 422 3 reaction with ammonia, 324, 326, 349 exitance flavonoid pigments, 323 32 radiant, 46, 372, 378 flavonol, 323, 325 spectral, 12 flavylium cation(s), 325, 330, 330, 331 explosives, detection of, 413 polymerization of, 331, 332 exposure meter, 471 flint glass, 60, 61 extinction, 34, 175 6 flower colours, 323 8, 329 see also attenuation fluorescein and derivatives, 367, 368, 378, 405 6, 409 extinction coefficient, 34, 51, 477 fluorescence, 34, 365, 366 extinction cross-section, 186 absorption mechanism for, 410, 411 extinction efficiency (factor), 186, 187 anti-Stokes, 394 extinction position, 147 compared with phosphorescence, 367, 369, 370 1 extraordinary light transmission, 487 molecular, 405 extraordinary ray (e-ray, E-ray), 138, 141, 142 fluorescence lifetime, 372, 378 eye fluorescence microscopy, 363, 409 10 colour of, 188 fluorescence quenching, 374 colour sensitivity, 10, 24, 25, 476 fluorescence resonance energy transfer, 376 7 Index 498

fluorescent brighteners, 368 gemstones, 149, 150 1, 213 18, 277 81, 283 4, 285, 286, 344 5 fluorescent centres, 381 irradiation of, 431 2 fluorescent coatings, in vapour lamps, 263 4, 383 see also amethyst; aquamarine; diamond; emerald; opal; ruby; fluorescent dyes, 405 6, 409, 456 sapphire; topaz fluorescent lamps, 379 83 geranium, colours, 328 Colour 80, 381 2 GFP (green fluorescent protein), 407, 408–9, 456 Colour 90, 382 3 Gladstone Dale formula, 61 2 colour temperature, 14 refractive coefficient, 61 2 halophosphate, 379 80 listed for various oxides, 63 phosphors in, 365, 379, 381, 382 3 glass trichromatic, 381 2 aluminoborosilicate, 483 fluorescent markers, 409, 412 chalcogenide, 80, 242 fluorescent molecules, 405 11 chemical impurities in, 80 1 fluorescent nanoparticles, 411 12 Co2 þ in, 285, 287, 288 fluorescent proteins, 407 8, 409 devitrified, 188 colours, 407 dichroic, 193 fluorescent sensors, 412 13 flint, 60, 61 fluorescent tags, 406 fluoride, 80, 287 fluorite, 366 lanthanide-containing, 190, 191 fluorochrome, 409 lead ‘crystal’, 60 fluorophore, 367, 409 metallic, 416 foam, 62 Nd3 þ in, 289 as antireflection coating, 108 opal, 42, 188 food wrap film, 148, 149 photochromic, 482 4 fool’s gold, 437, 438 photosensitive, 118 Forster€ distance, 376 polychromic, 481 2 Forster€ resonance energy transfer (FRET), 376 7, 410 Pyrex, 483 forward bias, 444 5, 472 ruby, 191 3 fovea (in eye), 24 second-harmonic generation in, 160 1 fracture damage, detection of, 416 selenide, 80 Fraunhofer diffraction, 198, 199 silicate, 226 Fraunhofer lines, 255 6 stained, 37, 194 Frenkel defects, 483, 484 structure, 287 Frenkel excitons, 423 4 window, 79, 119 frequency, 7, 44 ZBLAN, 80 angular, 44 glass ceramics, 188 relationship to velocity, 7 glass fibres, 77 temporal, 7, 44 attenuation in, 79, 80 frequency doubling, 115, 151, 153, 355 glaucoma, 162 see also second-harmonic generation glow stick, 413 14 frequency matching, 157 reactions in, 413 14, 414 frequency mixing, 155 6 glow-worm, 414 15 frequency trebling, 153 glycosides, 324, 329 see also third-harmonic generation gold frequency up-conversion, 394 colour, 478 Fresnel diffraction, 198 reflectivity, 477 Fresnel’s laws, 132 gold nanoparticles, 192, 479 80 Fritillary butterfly, 115 colours, 191 2 frontier molecular orbitals, 310 in Raman spectroscopy, 487 fuchsia, colours, 327 gold sols, 191 fuels, markers in, 354 5 graded-index materials, 64, 65 fused-fibre coupler, 56,57 disordered, 225 31 see also GRIN... gallium aluminium arsenide, 155, 448, 451 graphene, 10 gallium arsenide, 448 grating(s) gallium arsenide gallium phosphide alloys, 445 amplitude, 198, 206, 207 gallium nitride, with lanthanoids, 446, 447 blazed, 208 gallium nitride indium nitride system, 441, 442, 445, 455 chirped, 117 gallium phosphide, 445 diffraction: see main entry: diffraction grating gallium trioxide, 433 disordered, 225 31 gamut of colours, 31 fibre Bragg, 115 19 garnet, 286 Hill, 115 Garnet Star, 14 one-dimensional, 208 garnet structure, 286, 295 phase, 198 gas analysis, 353 4 reflection, 198, 205, 206, 207, 210 11, 232 gas discharge lamps, 256 9 three-dimensional, 211 25, 231 gas plasma display, 259, 260, 383 5 transmission, 198, 206, 207, 209 Geissler tube, 257 two-dimensional, 208 10, 230 1 gelatine, 242, 484 ultrahigh spatial-frequency, 109, 232 499 Index

uniform, 117 hydrogen spectrum, 249 51 see also diffraction grating hydrogen tungsten bronzes, 465, 466, 467 grating equation, 205 hyperchromic shift, 317 Gr€atzel cell, 472 hyperlenses, 87 9 green fluorescent protein (GFP), 407, 408–9, 456 hyperpolarizability, 161 GRIN antireflection coatings, 108 10, 121 hypochromic shift, 317 GRIN materials, 64, 65 hypsochromic shift, 317, 374 GRIN optical fibres, 82, 84 Grotrian diagram, 258, 258, 262, 264 IC (integrated circuit) manufacture, 106, 204 ground state, 4, 17, 249, 253 Iceland spar, 139, 140, 147 ground-state absorption (of photons), 369, 396 7, 399, 401 illuminance, 46 ground-state term, 306 ilmenite, 345 gypsum, 366 image: see holographic image; latent image Gyricon process, 39, 40 image reconstruction (of holograms), 235 6 IMOD displays, 110 11 haem, 320, 322, 323 impurity colours haematite, 322, 346 in diamond, 424 8, 446 haemocyanin, 323 in insulators, 424 haemoglobin, 322 impurity ion, and frequency doubling, 151 halite, 61, 138, 366 incandescence, 11, 363 halo, 75, 76 and colour, 11, 13 14 halogen vapours, colours, 311 12 spectrum, 11, 247 halophosphate fluorescent lamps, 379 80 incidence Hamburg blue, 342 angle of, 51, 52,92 Han blue, 297 normal, 92 Han purple, 297 plane of, 51, 92 HD-DVDs (high-definition digital versatile/video discs), 94, 448 incoherent light, 7, 11, 16, 17 heavy water, 316 incoherent scattering, 197 Heidinger’s brushes, 184n index of refraction: see refractive index Heisenberg uncertainty principle, 449 indicators, 350 3 helium, 256, 257 indicatrix, optical, 143, 144, 145 6 helium neon (He Ne) laser, 21, 211, 259 62, 436 indigo, 335 6, 336 high-brightness LEDs, 445, 445 indium oxide, 420, 441 high-reflectivity surfaces, 110 indium phosphide, 448 highest occupied molecular orbital (HOMO), 310, 455, 457, 458, 459 indium tin oxide (ITO), 439, 460, 464 see also HOMO LUMO separation indole-2,3-benzopyrrole, 378 Hill gratings, 115 inelastic scattering, 33 4, 175, 486 7 hole, 421 ‘inert’ gases effective mass, 451 colours in gas discharge lamps, 257 subbands (energy levels), 451, 453, 454 electron configurations, 261, 301 hole-excess centres, 430, 432 3 see also argon; helium; krypton; neon; xenon hologram(s), 235 43 infrared radiation, 10 amplitude, 241 insects Benton, 239 bioluminescence, 414 15 embossed, 242 3 detection of polarized light by, 184 and interference patterns, 235 eyes, 10, 64, 108, 109, 231 master, 239 wing markings, 91, 115, 188 phase, 241, 242 see also butterfly; firefly; moth planar, 237 insulators, 420 polarization, 241, 242 colours, 420 rainbow, 239 40, 242 impurity colours in, 424 recording media for, 240 2 intensity, 372, 395 reflection, 237 9 interaction energy, 276, 277 thick, 237 9 intercombination band, 279, 281 thin, 237, 242 interface transfer, 239, 240 reflection at, 92 4 transmission, 235 7, 239 refraction at, 54, 55 volume, 237 9 second-harmonic generation at, 161 holographic image, reconstruction of, 235 6, 237, 238 interference, 7 9 HOMO (highest occupied molecular orbital), 310, 455, 457, 458, 459 constructive, 8, 96, 97, 98, 228, 235 HOMO LUMO transitions, 317, 350, 357, 419, 468 destructive, 8, 9, 96, 99, 235 homochiral molecules, 166 of polarized light, 131, 148 ‘horse and rider’ double star, 204 at thin films, 94 9 HSB (hue/saturation/brightness) colour model, 28, 29 interference filters, 114 hue, 28 interference-modulated (IMOD) displays, 110 11 Hund’s rules, 306, 307 internal energy conversion, 369, 376 hydrangea, colour, 327 intersystem crossing (ISC), 369, 371 hydrogen bonding, 316 invisibility, 41, 62, 87, 135 hydrogen peroxide, 413, 414 ionizing radiation, F centres produced by, 429, 431 2 Index 500 iridescent colours, 91, 94, 122, 230 semiconductor diode, 155, 448, 449 iron compounds, colours, 284, 286 three-level, 281 2 iron-containing minerals, 149, 322, 345, 346 8, 437 titanium sapphire, 282 3 iron ions, detection of, 350, 351 type II behaviour, 434 iron pyrites, 437, 438, 439 laser cavity geometry, 21 2 irradiance, 46 laser cavity modes, 21 3 changes detected by eye, 92 laser light, interference observed using, 8 diffraction patterns, 199, 201 laser measuring equipment, 448 factors affecting, 154 laser pointer, 211, 448 irradiance profile, diffraction by a slit, 199 latent image, 435, 485 isomers: enantiomers lazurite, 348 isotropic substances, 54, 137, 138 LBO (lithium triborate), 153, 156 strain analysis, 148 LCD: see liquid crystal display ITO (indium tin oxide), 439, 460, 464 lead chromate, 346 lead crystal glass, 60 j j coupling, 253, 303 lead oxide, in flint glass, 60, 61 Jablonski diagrams, 371, 406 lead tungstate, 417 Japanese maple, leaf colours, 334 leaf colours, 321–2, 332, 333–4, 335 jellyfish, 62, 135, 407, 414, 415 leaf senescence, 332 jewellery, 110, 122, 425 LED: see light-emitting diode see also gemstones lemon yellow, 346 lens kaempferol, 323, 325 air, 64 KDP (potassium dihydrogen phosphate), 153 eye, 64, 190 keratin, 234 photochromic, 358 kinetics, luminescence, 370 1 super, 87 9, 204 King’s yellow, 437 leucoemeraldine, 468, 469, 469 Kitt Peak Observatory, 475 level, energy: see energy level Kroger Vink point defect notation, 426n, 440n lever rule, 31 LIDAR, 156 labradorescence, 122, 125 lifetime labradorite, 122, 124, 125 of excited states, 282 laevorotatory molecules, 164, 165, 166 fluorescence, 372, 378 Lambert’s law, 34, 176 of spectral holes, 299 lamp(s), 16 17 ligand-field splitting, 266, 268 fluorescent: see main entry: fluorescent lamps see also crystal field gas discharge, 257 9 light mercury-vapour, 247, 248, 263 4 absorption and emission of, 17, 18 sodium-vapour, 189 90, 247, 262 3 absorption of, 4 tungsten-filament, 14, 16 coherent, 7, 8, 17, 235 lanthanide, 190n, 301n diffraction of, 33 lanthanoid compounds, colours, 295 generation of, 10 13 lanthanoid-doped crystals, 297 9 incoherent, 7, 11, 16, 17 lanthanoid elements, 249, 254 interaction with materials, 33 6 electron configurations, 301, 303 monochromatic, 7 lanthanoid ions particle/corpuscular theory, 1, 2, 3 colours, 288 90 polarized: see main entry: polarized light electron configurations, 303, 382 reflection of, 33, 34 energy-level diagrams, 291, 292, 293, 381, 383, 390, 393, 447 scattering of, 33 4 in gallium nitride, 446, 447 unpolarized, 7, 11, 16, 129, 228 in glass, 190, 191 velocity in vacuum, 1 2, 7, 43 in insulators, 424, 425 wave theory, 1, 2, 3 in phosphors, 381 2, 385, 389, 392 4 light waves, 5 7 lapis lazuli, 348 and colour, 9 10 Laporte selection rule, 254 5, 270 interference of, 7 9 Large Hadron Collider, 417 polarization of, 129 35 large particles, scattering by, 184 7 light-emitting devices, 446 laser, 17 light-emitting diodes (LEDs) argon-ion, 115 active layer in, 445 carbon dioxide (CO2), 156 applications, 82, 173 colour centre, 434 5 blue, 445, 447, 448 continuous mode, 294, 358 depletion region in, 444 dye, 355 8 direct band gap materials, 442, 443 first demonstrated, 17, 21, 259, 281 displays using, 446 7 four-level, 290, 292 4 green, 445, 447, 454 helium neon (He Ne), 21, 211, 259 62, 436 heterojunction, 445, 446 neodymium (Nd:YAG, Nd:YLF), 156, 160, 290, 292 4 high-brightness, 445, 445 pulsed mode, 282, 294, 358 homojunction, 444, 445 ruby, 17, 21, 259, 276, 281 2 idealized structure, 443 5 501 Index

impurity doping in, 446 luminous paints, 365, 434 indirect band gap materials, 442 3, 443 lutein, 318, 319 organic: see main entry: organic light-emitting diodes lycopene, 317, 318 photometric characteristics, 45 Lycurgus Cup, 193 4 red, 445, 447 Lyman series, 251 transition region in, 444 white light generation, 447 8 magnesium aluminosilicate, 184 yellow, 445, 447 magnesium fluoride, 105, 106 lighting, 189 90, 247, 248, 262 4, 383, 447 magnesium oxide, 190, 384 lightness, 28 surface colour centre on, 434 limestone, 188 magnetic bubble memory, 474n limonene, 166 magnetic dipole transition, 254 line(s) magnetic field vector, 5, 6 Fraunhofer, 255 6 magnetite, 345 ‘persistent’, 257 malachite, 284 5, 286 sodium D, 263, 264 mallow, colours, 329 spectral, 248 Malus law, 137 telluric, 256 malvin, 328, 329, 331 line spectrum, 248, 249 51 manganese compounds, colours, 286 linewidth, natural, 248 manganese ions, in phosphors, 379, 380 liquid crystal, 168 73 maple, leaf colours, 334, 335 liquid-crystal display (LCD), 169 73 Marbled White butterfly, 324, 326 active matrix display, 39, 170 marine animals, 41, 62, 125, 135 light source in, 170, 173 marker reagents (for fuels), 354 5 passive display, 170 markers, fluorescent, 409, 412 liquid-crystal mesophases, 168, 169 masers, 21 calamitic, 168 mask chiral nematic, 228, 229 fibre Bragg grating, 118 cholesteric, 168, 228 30 integrated circuit, 106 columnar, 168 mass absorption coefficient, 35 director in, 168, 169, 228 mauveine, 337, 338 disclinations in, 168 Meadow Brown butterfly, 340 discotic, 168 melanins, 337, 339, 340 nematic, 136, 168, 169 melanocytes, 337 smectic, 168, 169 melanosomes, 337 twisted nematic, 168, 228 mercuric sulfide, 437, 438 liquid-crystal thermometer, 230, 420 mercury liquid scintillation counters, 417 energy-level diagram, 264 lithium compounds, colours, 255 ground-state configuration, 263 lithium iodate, 153 mercury-vapour lamps, 247, 248, 263 4, 383 lithium niobate, 153 replacement of mercury with xenon, 402 3 lithium triborate, 153 meso- form, 165 lithium tungsten bronzes, 466, 467 metal ions, detection of, 349 50 litmus, 350 1, 352 metal nanoparticle sensors, 486 lodestone, 345 metal oxide semiconductor (MOS) device, 475 longitudinal cavity modes, 22 metallic glass, 416 longpass filters, 114 metallic mirrors, 111, 478 Lorentz Lorenz equation, 59, 61 metallic nanoparticles, colours, 478 87 Lorenz Mie theory, 184n ‘metallic’ paints, 478 low-emissivity windows, 119 21 metalloanthocyanin, 327 8 low-reflectivity films, 105 10 metals lowest unoccupied molecular orbital (LUMO), 310, 455, 457, colours, 477 8 458, 459 reflectivity, 477 8 LS coupling: see Russell Saunders coupling metamaterials, 84 6 Lucalox, 190 metarhodopsins, 26, 27 luciferase, 415 metastable trapping, 299 luciferins, 415 methyl orange, 352 luminance, 28, 46 methyl red, 352 luminescence, 16, 363 418 mica, 147 cooperative, 394 microscopy early studies, 363 5 electron, 87 meaning of term, 363 fluorescence, 363, 409 types, 366 multiphoton excitation, 410 11 luminescent materials, 363 optical, 162 luminiferous aether, 2 polarizing, 168 luminous efficiency, 25, 443, 455 resolution of, 57, 87, 203 5 luminous exitance, 46 second-harmonic, 162, 163 luminous flux/power, 46 two-photon fluorescence, 411, 412 luminous intensity, 46, 367 microwave absorption and emission, 310 Index 502 microwaves, 2,16 n-doping, 464, 467 Mie Debye theory, 184n N517 dye, 474 Mie scattering theory, 184 7, 190, 194, 410, N719 dye, 474 478, 481 nacre (colour of shell), 122 Miller indices, 218n nanoparticle sensors, 486 minerals, 122, 124, 125, 149, 184 nanoparticles colour centres in, 430 1 in antireflection coating, 109 10 crystal-field colours, 284 5, 286 in coloured glass, 191, 193 fluorescent, 366 fluorescent, 411 12 iron-containing, 149, 322, 345, 346 8, 437 metallic, 478 87 mirages, 64 nanorods, 109 10, 450 mirror eye, 125 6 tuning of absorption characteristics, 480 mirrors nanostructures, 449 50 dielectric, 111 12 nanotubes, 136, 450 metallic, 111, 478 nanowires, 450 ‘smart’, 464 naphthopyrans, 359, 360 molar (decadic) attenuation coefficient, 36 National Ignition Facility (US fusion research), 295 molar refraction, 61 negative-index materials, 84 9 molecular crystals, excitons in, 424 metamaterials, 84 6 molecular fluorescence, 405 superlenses, 87 9 molecular orbital, 309 10 neodymium (Nd:YAG or Nd:YLF) lasers, 156, 160, 290, 292 4 antibonding, 310, 457 neon bonding, 310, 457 colour, 257 frontier, 310 energy levels, 260, 261 highest occupied, 310, 455, 457, 458, 459 Grotrian diagram, 258 lowest unoccupied, 310, 455, 457, 458, 459 line spectrum, 258 nonbonding, 310, 457 in plasma display, 384, 385 p, 310 ‘neon’ signs, 16, 257 p, 310 Nernst glowers, 16 molecular orbital theory, 310 net curtains, 209 molecular polarizability, 161 Newton’s black film, 100 molecule(s) nickel compounds chiral, 164 absorption spectrum, 264, 266, 271, 277, 278 dextrorotatory, 164, 165, 166 colours, 264, 265, 274, 277, 278, 286 donor p-bridge acceptor, 406, 410 detection of, 350 energy levels of, 309 11 Orgel diagram, 276 homochiral, 166 nickel oxide, hydrated, 467 laevorotatory, 164, 165, 166 NIMs: see negative-index materials organic: see main entry: organic molecules niobium pentoxide, 467 photochromic, 358 60 reduction of, 341, 467 as scattering centres, 178, 179 nitric oxide, formation in firefly, 415 molluscs, dye derived from, 336 nitrogen molecules, ionization of, 313 molybdenum blue, 341 noble gases: see ‘inert’ gases molybdenum trioxide, 100 nonbonding molecular orbitals, 310, 457 monochromatic light, 7 non-crossing rule, 276 monodisperse suspension, 218, 230 nonequilibrium state, 18 moon nonlinear crystals, 151 3, 154 5 colour, 180, 188 nonlinear effects, 151 7 eclipse, 180, 182 colour production by, 153 mordants, 337 nonlinear optical materials, 153, 155 MOS (metal oxide semiconductor) device, 475 nonlinear optics, 152 moth nonradiative transition, 279, 281, 282, 293,369,455,456 eyes, 64, 108, 109, 232 wing scales, 29 30 o-ray (ordinary ray), 139, 141, 142 mother-of-pearl, 122 oak, leaf colours, 332, 335 Mott Wannier excitons, 423 object beam (holograms), 235 Mount Palomar telescope, 204 octahedral coordination, 266 7, 268, 269, 270 1 mullite, 42 octahedral crystal field, 271, 272, 274 6, 282 multilayer stacks, 111, 113 14 oil film, 99, 104 disordered, 114, 115 OLEDs (organic light-emitting diodes), 459 64 tunable, 114 oligomers, 332 multiphoton absorption (of photons), 369, 370 olivine structure, 284, 286, 296 multiphoton excitation microscopy, 410 11 ommatidia, 108, 231 multiple quantum well (MQW) structure, 450, 451, 452 opal multiple scattering, 190 1 artificial, 218, 220 multiplexing, dense wavelength division, 119 Bragg equation for, 215 17 multiplicity selection rule, 270, 278 colloidal, 218 Munsell colour cylinder, 28, 29 colours, 214, 215, 216 Munsell colour solid, 28 common (potch opal), 213 503 Index

diffraction by, 213 18 Orion type stars, 14 fluorescence, 366 orpiment, 437 inverse, 218, 220 orthochromatic photographic film, 486 effective refractive index, 221 3 oscilloscope, 386, 389 precious, 214 17 overtones, in water molecules, 315, 316 total internal reflection in, 216, 217 oxidation processes, 463, 467, 468 opal glass, 42, 188 oxygen atoms, 314 opalescence, 214 opsin proteins, 24, 26 p-doping, 463, 467, 468 optic axis, 139, 140 p-orbitals, 300 optical activity, 162, 164 8 p-wave, 87, 131 optical band gap, 420 reflection of, 131 3, 134 listed for various oxides, 422 see also ray, extraordinary optical communications, 75, 77 paint, 188 9 optical constants, 51, 477 ‘metallic’, 478 optical density, 36 palladium compounds, detection of, 350 optical fibre(s), 77 8 panchromatic photographic film, 486 attenuation in, 79, 80 PANI: see polyaniline chemical impurities in, 80 1 Paris blue, 342 cladding of, 77 Paris green, 297 core, 77 parity selection rule, 254, 278, 279 coupler, 57, 57 particle detectors, 417 dispersion in, 81 2, 83 particle theory of light, 1, 2, 3 fibre-drawing process, 78 Paschen series, 251 graded-index, 82, 84 path difference, 96, 98, 126–7 monomode, 84, 115 paua shell, 122, 125 new materials, 80 Pauli exclusion principle, 300, 305 preform for, 78, 295 PEDOT: see poly(3,4-ethylenedioxythiophene) refractive index modulation in, 115, 117 pelargonium, colours, 329 repeaters, 294 peony, colours, 329 second-harmonic generation in, 160 1 period signal addition/removal from, 119 spatial, 44 signal amplification in, 294 5, 446 temporal, 44 stepped-index multimode, 82, 84 periodic table, 249 optical filtering, 38, 114, 224 permittivity, relative, 58 optical indicatrix pernigraniline, 468, 469, 469 biaxial crystal, 145 6 perovskite, 61 uniaxial crystal, 143, 144 perovskite bronzes, 465 optical masers, 21 perovskite type structure, 395, 465 see also lasers petunia, colours, 329 optical parametric amplifiers, 156 7 Pfund series, 251 optical parametric oscillators, 156 7, 355 pH indicators, 350 3 optical path, 53 pH sensor, 413 optical pumping, 156, 157, 282, 290, 292 pH theory of flower colours, 326 optical thickness, 53 phaeomelenin, 337 optically absent layer, 105 phase difference, 147 optically anisotropic materials, 138 phase grating, 198 optically isotropic substances, 54, 137 phase hologram, 241, 242 optically negative crystal, 143, 147 phase matching, 158 60 optically positive crystal, 143, 146 birefringent crystals, 160 orbital(s) phase object, 198 atomic, 264, 266, 267, 300 phase speed/velocity, 7, 44 d, 264, 266, 267, 300 phase of wave, 7 eg, 267, 269, 271 phenolphthalein, 352 molecular: see main entry: molecular orbital phonon absorption, 79 p, 300 phonon-assisted transition, 279 s, 300 phosphor electroluminescent displays, 391 4 t2g, 267, 269, 271 phosphorescence, 364, 366, 367, 370 ordinary ray (o-ray, O-ray), 138, 141, 142 compared with fluorescence, 367, 369, 370 1 organic electroluminescence, 457 9, 460 phosphors organic light-emitting diodes (OLEDs), 459 64 in cathode-ray television, 387 9 organic materials, second-harmonic and sum-frequency generation in fluorescent tubes, 363, 364, 365, 379, 381, 382 3 by, 161 2 photostimulable, 435 6 organic molecules photobleaching, 410 as insulators, 457 photochromic bleaching, 358, 484 interaction with light, 136 photochromic glass, 482 4 photochromic, 358 60 photochromic organic compounds, 358 60 organic semiconductor, 340, 457 64 photochromic plastics, 359 Orgel diagram, 274, 275, 276 photochromic reactions, 24, 26, 28, 359 Index 504 photochromic sunglasses and ski goggles, 359 and refractive index, 58 60 photoconductive effect, 471 polarization photodiode, 471 colour produced by, 148 photoelectric effect, 3, 14 15 linear, 6, 7, 130 1 photoelectrochemical cells, 472 and optical activity, 166 8 photoelectrons, 2 and phase matching, 159 photographic film, 484 6 by reflection, 131 5 black-and-white, 486 and scattering, 181 4 colour, 486 polarization hologram, 241, 242 orthochromatic, 486 polarized light panchromatic, 486 circular, 130, 131 photography colour affected by, 279 80 digital, 170, 172, 474 7 detection of, 137, 184 film, 241 2, 484 6 elliptical, 130, 131 photoionization, gated, 299 interference of, 131, 148 photoluminescence, 366, 446 plane (linear), 7, 130 atomic processes in, 368 78 rotation of, 162, 164 6 in quantum dots, 455 polarizer, 136 photometric units, 45, 46, 476 dichroic sheet, 136 photon(s), 3, 14 16 Polaroid sheet/sunglasses, 136, 148, 168, 184 absorption of, 369 polaron, 458 conversion, 369 polars, 135 7 emission of, 369 crossed, 137, 148 energy, 4, 15 sheet form, 136, 151 interaction with electron, 249 in tandem, 136 photon cascade emission, 403 poling, 161 photonic band gap (PBG), 223 4 thermal, 161 photonic crystals, 85, 218, 220 polyaniline (PANI), 468, 469 photonic engineering, 111 electrochromic device using, 470 in nature, 121 6 polychromic glass, 481 2 photonic stopband, 115, 223 colours, 482 photoprotein, 416 polychromic materials, 468 photoreceptor cells polycrystalline materials, 160 cones, 24 poly(3,4-ethylenedioxythiophene) (PEDOT), 469, 470 L (red) cone receptors, 24 electrochromic device using, 470 M (green) cone receptors, 24 polymer films, 148, 149, 151 S (blue) cone receptors, 24 polymers, electrochromic, 468 70 0 rods, 24, 26 poly(2-methoxy-5,2 -ethylhexyloxy)-1,4-phenylenevinylene (MEH- photorefractive materials, 60 PPV), 459, 461 photoresist, 106, 205n, 242 poly(methyl methacrylate) (PMMA), 218, 220 photosensitive materials, 118, 240, 241 2 polypyrrole, 468 photostimulable phosphors, 435 6 polysulfides, 348 9 photovoltage, 472 polythiophene, 468, 470 photovoltaic effect, 471 alkoxy-substituted, 468, 469, 470 photovoltaic materials, 471 poly(vinyl alcohol) (PVA), 136 photovoltaic solar cells, 471 2 population inversion, 18, 255, 259, 281, 283, 292, 448 phthalocyanines, 322 3 porcelain, 42, 188 Pigment Blue 15, 322 porous coatings, 108 pigments, 295 7, 322, 333 40, 437 porous materials, 62 pixels, 39, 169, 170 porphyrins, 319 22 Planck constant, 11, 13, 15, 43 positron emission tomography, 417 Planck law of black-body radiation, 11 13, 15, 20, 44, 45 potassium dichromate, 346 plasma, 259 potassium dihydrogen phosphate (KDP), 153, 154 5 plasma displays, 259, 383 5, 403 potassium iron(III) cyanide, 342, 343, 344 plasma frequency, 479 potassium permanganate, 345 plasmon, 478 9 potential, built-in/contact, 444, 471 surface: see main entry: surface plasmon potential well (in quantum structures), 451, 454 plasmon hybridization, 480 praseodymium ions plasmonic crystals, 488 colours, 289 plastic films, 148, 149, 151 energy levels, 447 plastics, appearance, 42 in quantum cutting, 403, 404 platinum, in glass, 192 up-conversion and, 401, 403 pleochroism, 149, 151 primary colours point defects, 395, 424, 440, 483 additive, 30 pointillism/pointillist painting, 29, 122, 387 subtractive, 37, 486 polariton, 479n principal refractive indices, 138, 140 polarizability principle of superposition, 7 8 electronic, 58 prism, spectrum formed by, 67 8, 139 molecular, 161 proanthocyanidins, 337 505 Index propagation number, 45 rare earth elements, 301 propagation vector, 5 see also lanthanoids proteins, folding and coiling of, 407 ray Prussian blue, 341 2, 344, 349, 467 8 extraordinary (e-ray, E-ray), 139, 141, 142 Prussian green, 342, 343 ordinary (o-ray, O-ray), 139, 141, 142 Prussian white, 342, 343, 344, 467, 468 Rayleigh criterion (for resolution), 203, 204 puddle, oil film on, 99, 104 Rayleigh Gans theory, 184 PVA (polyvinyl alcohol), 136 Rayleigh radiation, 487 pyramid, truncated, 445, 446 Rayleigh scattering, 177 8, 184, 185, 486 pyran, 359 in biological tissues, 190 pyrite, 437, 438 effects, 180, 188, 190, 410 in optical fibres, 79, 81 quality factor, 21 and wavelength, 179 quantum/quanta, 13 Rayleigh scattering pattern (polar diagram), 178, 182 quantum computers, 428 rays of light, 1, 3 quantum cutting, 402 4, 405 reaction quantum dots, 350, 351, 409, 450, 455 7 bimolecular, 378 colours, 455, 456 electrochromic, 464, 466, 467, 468 quantum optics/electrodynamics, 3 photochromic, 24, 26, 28, 358 9 quantum wells, 450, 451 4 reaction rate energy levels in, 452, 453 fluorescence and phosphorescence, 372 3 multiple, 450, 451, 452 photochromic glass, 484 quantum wires, 450, 454, 455 red-hot object, radiation from, 11, 12 quantum yield, 371, 372 red shift, 316, 374 quarter-wave stack, 111 12 red sunset, 179 quartz, 165 red wine, 328 32 smoky, 431 reduction processes, 463, 467 quenching, 374 8 reference beam (holograms), 235 concentration, 377 8 reflectance, surface, 92, 132 by defects, 369 reflection, 33, 34 dynamic, 374, 378 angle of, 92 by energy transfer, 376 7 coefficient of, 92 fluorescence, 374 colour production by, 91 128 by molecular collisions, 369 data storage using, 94 static, 374 diffuse, 42 thermal, 375 6 from transparent plate, 92 4 quercetin, 323, 325 perpendicular to film, 96 7 quinizarin, 354, 355 polarization by, 131 5 total internal: see main entry: total internal reflection racemic acid, 162, 164 reflection diffraction gratings, 198, 205, 206, 207, 210 11, 232 racemic mixtures, 164, 165 reflection holograms, 237 9 radar, 386 reflectivity radar backscattering efficiency, 194, 195 high, 110 radiance, 46 of metals, 111, 477 8 radiant exitance, 46, 372, 378 surface, 92, 93 4, 132 radiant flux/power, 46 of thin film in air, 101 2 radiant intensity, 46 refraction, 33, 49, 50 radiation angle of, 51, 52 absorption of, 17, 18 colour production by, 67 75 emission of, 17, 18 double: see main entry: double refraction radiationless transition, 279, 281, 282, 293, 369 at interface, 54, 55 radiative transition, 369 molar, 61 radio waves, 2,16 specific, 61 radioactivity, 365 refractive coefficient, 61 2 recording of, 417 listed for various oxides, 63 radioluminescence, 365, 366 refractive index, 49, 51 radiometric units, 12 13, 45, 46, 476 absolute, 52 radium, 365 average, 62, 108 rainbow, 68 75 complex, 51, 191, 477 deviation of rays, 69, 70, 73, 74,75 and crystal structure, 140 3 impact parameter, 69, 70, 73, 74,75 and density, 60 2, 138 polarized, 75 effective, 221 primary, 68, 69 71, 69, 71, 72, 73 of inverse opals, 221 3 secondary, 68, 69, 71 2, 71, 74,75 of foam, 62 ternary and high-order, 75 graded, 64 rainbow holograms, 239 40, 242 listed for various substances, 61 raindrops, reflection within, 69, 72, 73–4, 134, 135 of metals, 477 Raman effect, 487 of mixtures, 62 Raman spectroscopy, 486 7 negative, 84 Index 506 refractive index (Continued) Scheele’s green, 297 nonlinear, 54 scheelite, 366 and polarizability, 58 60 schiller, 122, 124, 125 of porous materials, 62 schlera (in eye), 190 principal, 138, 140 schorl, 149 and symmetry, 137 9 scintillators, 365, 416 17 and wavelength, 52 4, 102, 158, 178 properties required, 417 refractive index grating, 115 second-harmonic generation (SHG), 151, 153 4, relative permittivity, 58 155, 156 resolution limit, of optical instruments, 57, 87, 203 5 colours produced by, 154 5, 160 resonance, 22, 479 at interfaces, 161 resonant condition, 376 microscopy using, 162, 163 resonant frequency, 376 in organic materials, 161 2 retardation, 96 in polycrystalline materials, 160 and colours, 126–7 selection rules relative, 147 Laporte rule, 254 5, 270 retina, 24, 125, 162, 163 multiplicity rule, 270, 278 retinal, 24, 26 parity rule, 254, 278, 279 11-cis-retinal (retinal1), 26, 27 selenium exposure meter, 471 all-trans-retinal, 26 self-cleaning windows, 121 all-trans-retinal rhodopsin, 26, 27 self-quenching, 378 RGB colour model, 30 Sellmeier constant, 66 rhinestones, 110 Sellmeier equation, 66 rhodamine 6G dye, 356 semiconductor rhodopsin, 24, 26, 27,28 colours, 436 9 ring silicate pigments, 297 degenerate, 440 rods and cones: see photoreceptors extrinsic, 436n rose, colours, 327 inorganic, 436 41 rosemary, 321 intrinsic, 436 rotational energy levels, 310, 311 isostructural pairs, 441 roughness, surface, 40, 42 organic, 340, 457 64 rubrene, 414, 415 see also transparent conducting oxide ruby, 150, 277 semiconductor alloys, colours, 440 colour of, 150, 277 81, 286 semiconductor diode lasers, 155, 294, 448, 449 dichroism in, 150 1, 279 81 semiconductor LED, 443 ruby glass, 191 3 semiconductor nanostructures, 449 50, 449 57 ruby laser, 17, 21, 259, 276, 281 2 sensitizer, 366, 367, 472, 473 4 Russell Saunders coupling, 252, 253, 302 SERS (surface enhanced Raman spectroscopy), rutile, 121, 138 9, 432 3 486 7 Ryberg constant, 250 SFG (sum frequency generation), 155, 156 SFM (sum frequency mixing), 155 s-orbitals, 300 shells, colour of, 122 s-wave, 131 SHG: see second-harmonic generation reflection of, 131 3, 134 shims (for embossed holograms) see also ray, ordinary child/stamper, 242 3 saffron, colour, 319 mother/master, 242, 243 sage, 322 shortpass filters, 114 St Elmo’s fire, 259, 315 SI units, 43, 46 sapphire, 345 see also main entry: units see also titanium sapphire laser SiAlONs, 43 saturation, 28, 31 signal beam, 235 scallop, eye, 125 silica optical fibres, 77 8, 84 scanning electron microscopy, 390 chemical impurities in, 81 scattering silica spheres, in opal, 214 15, 216 coherent, 197 silicon, band gap, 436, 437 colour production by, 175 96 silicon carbide, 103 elastic, 33, 175 silicon dioxide, 62, 103, 121 incoherent, 197 see also quartz; silica inelastic, 33 4, 175, 486 7 silicon oxynitride, 107 meaning of term, 175 silicon photovoltaic cell, 471, 472 multiple, 190 1 silver and polarization, 181 4 colour, 478 subsurface, 28 in glass, 192, 483 4 and transparency, 42 reflectivity, 111, 477 see also Mie scattering; Raman effect; Rayleigh scattering; silver gallium selenide (AGSe), 153 Tyndall scattering silver gallium sulfide (AGS), 153 scattering coefficient, linear (Napierian), 35, 176 silver halides scattering efficiency (factor), 186, 187 in photochromic glass, 483 scattering length, 176 in photographic film, 484 5 507 Index silver nanoparticles, 479 80 spin orbit coupling, 253, 277, 302 3, 306, 371, 381, 428 in polychromic glass, 481 2 spin quantum numbers, 304 in Raman spectroscopy, 487 spinels, 287, 295 6, 296 7, 345 single quantum well (SQW), 450, 451, 452 spiro-naphthoxazines, 359, 360 singlet states, 303, 371, 458, 459 spontaneous emission, 17, 18 ski goggles, 359 Einstein coefficient for, 19 sky spot test, 350 colour, 23, 40, 178 9, 180 SQW (single quantum well), 450, 451, 452 polarization of light from, 183 4 stained glass, 37, 194 small particles, scattering by, 184, 185 stars ‘smart’ mirrors, 464 and diffraction limit, 204 ‘smart’ windows, 119 21, 464 effective temperatures, 14 smoky quartz, 431 spectra, 256 Snel’s law (Snell’s law), 51 stepped-index multimode optical fibres, 82, 84 applications, 67, 87, 216, 218, 445 Stern Volmer constant, 378, 379 soap film, 91, 99, 100 1, 100 Stern Volmer equation, 378 sodium stimulated emission, 17, 18, 19, 21, 22, 259, 260, 282, 292, 369 Grotrian diagram, 262 Stokes radiation, 487 ground-state configuration, 263 Stokes shift, 365 line spectrum, 262 stopband, photonic, 115 sodium D lines, 263, 264 strawberry tree, 321 sodium racemate, 162, 164 street lighting, 189 90, 247, 248, 262 4, 383 sodium tartrate, 162, 164 stress, 148 sodium tungsten bronzes, 466 stress birefringence, 148 sodium vapour lamps, 189 90, 247, 262 3 stretching modes solar cells, 471 2 antisymmetrical mode, 315 dye-sensitized, 472 4 symmetrical mode, 315 solar concentrators, 472 strontium aluminate phosphor, 433 4 sols, 191 3, 478 strontium compounds, colours, 255, 315 solvatochromic fluorophore, 409 strontium magnesium phosphate, 383 solvatochromism, 374, 406 strontium nitrosilicide phosphor, 447 Solvent Yellow, 354 5, 356 structural interactions, in luminescence, 374 sonoluminescence, 315 structural probe, colour as, 287 space charge, 444, 471 subpixels, 170 spatial period, 44 subtractive coloration, 37 9, 194, 486 see also wavelength sum frequency generation (SFG), 155, 156, 161 specific refraction, 61 sum frequency mixing (SFM), 155 specific rotation, 164 sun specific rotation dispersion, 167 colours, 179, 180, 188 spectral exitance, 12 effective temperature, 14 spectral hole radiation from, 11, 12, 179, 180, 313 homogenous linewidth, 298 spectrum, 255 6 inhomogenous linewidth, 298 sun bed tubes, 383 lifetime, 299 sun tan, 337 spectral-hole burning, 297 9 sunglasses, 136, 148, 359 mechanisms, 299 sunscreens, 51, 190, 346, 420 method, 298 sunset, 179, 180 spectral-hole formation, 297 300 sunstone, 184 spectral irradiance, 12, 46 superlenses, 87 9, 204 spectral lines, 248 surface colour centres, 434 spectral radiance, 11 surface enhanced Raman spectroscopy (SERS), 486 7 spectrometer, 247 surface plasmon, 479 spectroscope, 247n, 255 surface plasmon polaritons, 479 spectrum energy levels, 480, 481 absorption: see main entry: absorption spectrum localized, 480 of atoms, 247 51, 254 5 longitudinal, 480, 481 band, 312, 313 propagating, 480 continuous, 11, 247 transverse, 480, 481 electromagnetic, 2, 247 surface plasmon resonance, 479 emission: see main entry: emission spectrum surface plasmon resonance spectroscopy, 480 formation of, 67 75 surface reflectivity, 92, 93 4, 132 of ions, 247 51 surface roughness, 40, 42 line, 247 8 symmetry operators, 137 solar, 255 6 stellar, 256 talc, 366 visible, 2, 9, 10, 10 tannins, 332, 337 sphalerite, 366 tantalum nitride, 437 spin-allowed transitions, 270, 276, 281 tartaric acid, 162, 165, 166 spin-forbidden transitions, 278 meso-form, 165 Index 508

TE (transverse electric) wave, 131, 132, 133 in LEDs, 445, 446 telescopes, 204, 478 in opals, 216, 217 television sets, 170, 384, 386 9 tourmaline, 149 50, 286 telluric lines, 256 transition TEM (transverse electromagnetic) waves, 5 6 allowed, 270 temperature sensor, 413 charge-transfer, 340, 342, 345 6, 348 9 temporal frequency, 7, 44 electric dipole, 254 temporal period, 44 electron pair, 348 tenebrescence, 366 electronic, 258, 262, 263 4, 275, 280, 281 terbium ions forbidden, 270, 278, 281 energy levels, 381, 393 HOMO LUMO, 317, 350, 357, 419, 468 in phosphors, 381 2, 393 interband, 451 4 in quantum cutting, 403 4, 405 intersubband, 453, 454 term (of atom or ion), 251, 252, 303 laser, 260 1 multiplicity of, 252, 271 magnetic dipole, 254 term splitting, 271 3 ntop, 310 term symbol, 252, 303 nonradiative, 279, 281, 282, 293, 369, 455, 456 tetrahedral coordination, 268, 269 p to p , 310, 322, 468 tetrahedral crystal field, 271, 272 3 parity-forbidden, 281 TFEL (thin-film electroluminescent) displays, 391 4 phonon-assisted, 279 thermal poling, 161 radiationless, 279, 281, 282, 293, 369 thermal quenching, 375 6 radiative, 369 thermochromic materials, 270, 360 rates, 281 thermochromism, 230, 270, 420, 437, 441, 470, 478 spectroscopic, 274 thermoluminescence, 366 spin-allowed, 270, 276, 281 THG (third-harmonic generation), 153, 154, 155, 156 spin-forbidden, 278 thin film(s) vibrational, 310 anodized, 103 4 transition metal compounds colour of colours, 284 5, 286, 295 7 in air, 99 102 pigments, 295 7 on substrate, 102 4 transition metal elements, 249, 301 interference at, 94 9 transition metal ion reflected beams, 96 7 colours, 264 70 transmitted beams, 98 9 crystal-field splitting, 270 7 multiple, 111 15, 121 2 electron configurations, 302 OLED, 459 60 in glass, 196 reflectivity of, 104 5 in insulators, 424, 425 on substrate transition-metal-ion lasers, 281 3 colour of, 102 4 see also ruby laser; titanium sapphire laser reflectivity of, 104 5 translucency, 42 tapered/wedge-shaped, 96 7 transmission diffraction gratings, 198, 206, 207, 209 thin layer, properties, 450 transmission electron microscopy, 390 thin-film coatings transmission holograms, 235 7 antireflective, 105 10 transmissivity, 36 high-reflectivity, 110 transmittance, 36 thin-film electroluminescent (TFEL) displays, 391 4 transparency, 34, 41 thin-film engineering, 111 transparent animals, 41, 62, 190 thinning film, 100 1 transparent ceramics, 43, 189 90 thiophene, 470 transparent conducting oxides (TCOs), 103, 437, third-harmonic generation (THG), 153, 154, 155, 156 437 8 thulium ions, 393 transparent insulating solids, 91 colours, 289 transparent materials, 34, 42, 43 energy levels, 420, 447 transparent plate, reflection from, 92 4 thymol blue, 352 transparent solids, 41 3 tin oxide, 119 20, 440 transverse electric wave (TM wave), 131, 132, 133 titanium carbide, 437 transverse electromagnetic (TEM) wave, 5 6 titanium carbonitride, 441 transverse magnetic wave (TM wave), 87, 131, 132, 133 titanium compounds, colours, 286 trapping, metastable, 299 titanium dioxide, 39, 62, 121, 138 9, 188, 190, 346 triboluminescence, 366, 416 reduction of, 341 trichroism, 149, 284 titanium nitride, 437 trichromatic fluorescent lamps, 381 2 titanium oxynitride, 437 trichromaticity, 24 titanium sapphire laser, 282 3 tridymite, 433 TM (transverse magnetic) wave, 87, 131, 132, 133 triiodide iodide redox couple, 474 TNT (2,4,6-trinitrotoluene), detection of, 413 triplet states, 304, 371, 458, 459 tomato, colour, 317 tungsten blue, 341 topaz, 286432 tungsten bronzes, 465 7 total internal reflection, 54 7, 77 tungsten carbide, 437 frustrated, 56,57 tungsten-filament lamps, 11, 14, 16 509 Index tungsten trioxide incoherent, 7, 11, 16, 17 electrochromic film, 465 7 monochromatic, 235 reflectance spectrum, 420, 421 progressive, 5 tuning, optical parametric oscillator, 157 propagating, 5 Turnbull’s blue, 341, 342 standing (non-travelling), 44 turquoise, 286 transverse electric, 131, 132, 133 two-frequency up-conversion, 401 transverse electromagnetic, 5 6 two-photon fluorescence, absorption mechanism for, 410 11, 411 transverse magnetic, 87, 131, 132, 133 two-photon fluorescence microscopy, 411, 412 travelling, 5, 44 Tyndall blue, 176, 187, 188 wave amplitude, 7 Tyndall scattering, 176, 187 wave equation, 6, 43 4 Tyndall spectra, 185 wave particle duality, 15 16 Tyrian purple, 336 7 wave theory of light, 1, 2, 3 wave vector, 5 ultramarine (pigment), 348 9 wave velocity, 5, 7 artificial, 349 relationship to frequency, 7 ultraviolet radiation, 10, 118, 364 wavelength, 7, 44 in IC manufacture, 106, 204 and energy, 44 5 in sun beds, 383 estimation by diffraction, 210 11 uniaxial crystal and Rayleigh scattering, 179 dichroism in, 149 and refractive index, 52 4, 102, 158, 178 double refraction in, 143 4 visible spectrum, 10 unit cell of crystal, 138 wavelength dispersion, 82 units wavenumber, 45 energy, 43 Whewell Quetalet diffraction pattern, 227 photometric, 45, 46 white-hot object, 11 radiometric, 12 13, 45, 46 white light spectral, 10 generation by LEDs/OLEDs, 447 8, 463 4 up-conversion, 151, 355, 369, 370, 394 402 perception of, 14 absorption mechanism for, 410, 411 refraction by prism, 67 8 two-frequency, 401 standard daylight, 31 up-conversion efficiency, 394 Wien displacement law, 13 uranium compounds willemite, 366 colours, 295, 296 window glass, 79, 119 radioactivity, 365, 431 windows low-emissivity, 119 21 vacancy, 424 self-cleaning, 121 vacuum, black, 257 ‘smart’, 119 21 valence band, 4, 5, 388, 419, 420 windscreen, 148 vanadium compounds, colours, 286 wine colour, 328 32 vector, 5 woad, 335 vector model of atom, 302 4 wollastonite, 366 velocity vector, 5, 6 work function, 3 vermilion, 437 working electrode (in solar cell), 473 vibration ellipse, 131 wurtzite, 366, 440 vibrational energy levels, 310, 311 vibrational transition, 310 X-ray diffraction viewing angle Bragg’s law, 212 13 rainbow, 68, 70, 71, 71,72 dynamical theory, 213 thin film, 97 8 kinematical theory, 213 Virginia creeper, leaf colours, 333, 335 ‘powder’ method, 231 visible spectrum, 2,9 X-ray imaging, 435 6 vision, 23 8 X-ray tomography, 417 visual pigments, 24, 26 8 xanthophyll, 318, 319 visual purple, 24 xenon, 257 vitamin C, 166 line spectrum, 258 volatile organic compounds (VOCs), detection of, 354 in plasma display, 384, 385 replacement in mercury-vapour lamps, 402 3 walk off, 159 xenon arc lamp, 257 water xenon flashlamp, 257, 295 colour of, 315 16 xenon flashtube, 257, 281 heavy, 316 refractive index, 70 ytterbium ions, energy transfer processes, 399 401 water air surface, reflection at, 134 5 yttrium aluminium garnet (YAG), 292, 447 water-lily, 340, 377 yttrium gadolinium borate, 385 wave yttrium oxide, in fluorescent lamps, 381 coherent, 7, 8, 17 evanescent, 54, 56, 57, 87, 89 ZBLAN glasses, 80 idler, 156, 157 zinc cadmium sulfide, silver-activated, 387, 389 Index 510 zinc oxide, 346, 420, 421 fluorescence, 364, 455 reflectance spectrum, 420, 421 with manganese ions, 391 zinc oxide nanoparticles, band gap, 422, 449 minerals, 366 zinc selenide, 455 silver-activated, 387 9 zinc silicate, 385 zincite, 366 zinc sulfide zircon, 366 crystal structures, 441 zirconium carbonitride, 441 with cuprous chloride, 389 zirconium nitride, 437